Understanding the effects of symmetric salt on the structure of a planar dipolar polymer brush

Mahalik, Jyoti P.; Sumpter, Bobby G.; Kumar, Rajeev

doi:10.1063/1.5037077

The effects of added salt on a planar dipolar polymer brush immersed in a polar solvent are studied using a field theoretic approach. The field theory developed in this work provides a unified framework for capturing effects of the inhomogeneous dielectric function, translational entropy of ions, crowding due to finite sized ions, ionic size asymmetry, and ion solvation. In this paper, we use the theory to study the effects of ion sizes, their concentration, and ion-solvation on the polymer segment density profiles of a dipolar brush immersed in a solution containing symmetric salt ions. The interplay of crowding effects, translational entropy, and ion solvation is shown to exhibit either an increase or decrease in the brush height. Translational entropy and crowding effects due to finite sizes of the ions tend to cause expansion of the brush as well as uniform distribution of the ions. By contrast, ion-solvation effects, which tend to be stronger for smaller ions, are shown to cause shrinkage of the brush and inhomogeneous distribution of the ions.

I. INTRODUCTION

Polar polymers such as poly(N-isopropylacrylamide) (PNIPAM), poly(ethylene oxide) (PEO), and poly(methyl methacrylate) (PMMA) have shown a great potential for a wide variety of biomedical^1–17 and bioanalytical^8,18–21 applications such as drug delivery, anti-biofouling surfaces, membranes, biologically active thin films, separation techniques, and sensing devices. Effects of inorganic salt on the structure and dynamics of the polar polymers^22–35 have been studied extensively due to their relevance for almost all biological systems. It has been shown that salt affects the lower critical solution temperature (LCST), above which a homogeneous solution containing the polar polymers turns cloudy due to phase segregation.³⁶ In particular, it has been reported that the LCST of PNIPAM solutions can be either decreased or increased by adding salt and the LCST is found to be sensitive to the chemical nature of the anion of the added salt. Furthermore, a shift in the LCST due to the added salt is found to follow the Hoffmeister series,²⁴ which was originally developed to classify ions as per their tendency to precipitate proteins dissolved in water. It has been postulated that interactions of anions with solvents such as water are responsible for the origin of the Hoffmeister series. In fact, anions have been categorized based on the nature of their interactions with the medium so that kosmotropes and chaotropes represent solutes which “make”/“stabilize” or “break”/“destabilize” existing water or solvent structure in the solution, respectively. Furthermore, kosmotropes such as F⁻, Cl⁻, and Br⁻ cause a decrease in the LCST and chaotropes (e.g., ${N O}_{3}^{-}$ ⁠, I⁻, and SCN⁻) tend to increase the LCST. However, recent studies focusing on the effect of solutes on the water molecules cast doubt on such a mechanism behind the Hoffmeister series.^37–39 It has been suggested that dispersion interactions may be playing an important role behind the applicability of the Hoffmeister series in polar polymers. Besides polymer solutions, polar polymer gels and brushes are also reported to follow the Hoffmeister series. For example, PNIPAM hydrogels have been reported to follow the Hoffmeister series,^22,26 whereas PEO gels do not follow such a series.²³ This indicates that the Hoffmeister series may be polymer specific. The effect of salt on the polymer gels is manifested in the form of swelling or deswelling with respect to a pure solvent medium. PNIPAM brushes are also reported to follow the Hoffmeister anion series,^{25,29,30,35,40} which manifest in the form of shrinkage or expansion of the brushes with respect to a pure solvent medium.

There are very few theoretical studies on understanding the effects of salt on the LCST behavior of polar polymers.^32–34 Using classical molecular dynamics, Du and co-workers investigated the effects of ion specificity on the interactions between PNIPAM and alkali salts with chloride anions.³² They showed that alkali metal cations bind strongly with the oxygen (O) in the amide group, and the size of ion determines the binding strength. However the divalent cations such as magnesium (Mg²⁺) and calcium (Ca²⁺) bind weakly with the oxygen and exhibit stronger solvation by the solvent. The stronger cation-amide interactions lead to an increase in the LCST while strong cation-water interactions cause a decrease in the LCST, which gets manifested as salting-in and salting-out, respectively, of PNIPAM. The divalent ions exhibited the salting-out effect due to the dominance of solvation by the solvent. Using a thermodynamic analysis, Heyda and Dzubiella³⁴ modeled the effects of kosmotropic and chaotropic salts on the LCST of PNIPAM. They showed that linear free energy changes based on generic excluded-volume mechanisms can explain the effects of strongly hydrated kosmotropes. Furthermore, the effects of less hydrated chaotropes, which tend to bind with the amide groups and exhibit non-monotonic effects on the LCST as a function of their concentration, can be understood by including competitive effects of preferential interactions of water as well as salt ions with PNIPAM. These theoretical studies have pointed out the importance of ion solvation, monomer-ion binding, and monomer-solvent interactions in affecting the LCST. In the case of inhomogeneous systems such as polymer brushes, partitioning of the salt ions and solvent molecules between the interior and exterior of the brushes along with self-consistent changes in polymer chain conformations needs to be taken into account. In this work, we develop and apply such a self-consistent theoretical framework to study the effect of added salt with symmetric ions on a polymer brush immersed in a polar solvent.

The theoretical framework presented here builds on our previous studies related to charge regulating polyelectrolyte brushes,^41,42 dipolar polymer blends,⁴³ and salt-free dipolar polymer brushes.^44,45 The framework provides a unified description of local as well as non-local polarization (or dielectric) effects and allows a self-consistent treatment of the ionic solvation. The theoretical framework is developed using a field theory approach, in which the effects of electrostatics are captured by going beyond the standard saddle-point approximation. We ignore the effects of ion binding and study the effects due to finite sizes of symmetric salt ions, ion concentrations in the electrolyte solutions, ion valencies, and dipolar interactions on a planar polymer brush. We note that the framework can be generalized in a straightforward manner to study the effects of applied electric field, asymmetric ions, and ion binding, however, at the cost of solving two extra numerically challenging equations.

This paper is organized as follows: the general formalism is presented in Sec. II and details about the application of the formalism to salty dipolar brushes are presented in Sec. III. Numerical results are presented in Sec. IV, and the conclusions are presented in Sec. V.

II. THEORY

We consider a planar polymer brush formed by n_p mono-disperse flexible chains (such as PNIPAM), each having N Kuhn segments of length b. The chains are assumed to be uniformly grafted onto an uncharged substrate so that the grafting density (defined as the number of chains per square nanometer) is σ (see Fig. 1). For the field theoretical analysis^46,47 described in this work, each chain is represented by a continuous curve of length Nb, and an arc variable t is used to represent any segment along the backbone so that t ∈ [0, N]. t = 0 corresponds to the grafted end, and t = N represents the free end. To keep a track of different grafted chains, a subscript α is used so that t_α represents the contour variable along the backbone of the αth chain. We use the notation R_α(t_α) to represent the position vector for a particular segment, t_α, along the αth chain. In particular, R_α(t_α = 0) = r_α(0) is the position vector for the grafted end of the αth chain.

FIG. 1.

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Schematic of a planar dipolar brush containing monodisperse polymer chains immersed in a salty solution. The dipolar chains are modeled as continuous curves (blue colored) containing electric dipoles (blue arrows), and the solvent molecules are represented as discrete electric dipoles (orange arrows). Cations and anions are represented as white beads and red beads, respectively.

Each segment along a chain is modeled as a dipole with a “smeared” spatial density distribution of width a_p.^44,45,48,49 The density distribution is represented by a function $ĥ_{p}$ (r − R_α(t_α)) centered at the center of mass of the segment (R_α(t_α)). Specifically, for each segment t_α along the αth chain, an electric dipole of moment (in units of electronic charge, e) p_α(t_α) = p_pu_α(t_α) is assigned, where u_α(t_α) is a unit vector specifying the direction of the moment and p_p is the magnitude of the moment. Similarly, each of the n_s solvent molecules is assigned an electric dipole moment and we use the notation p_k = p_su_k to represent the dipole moment of the kth solvent molecule oriented along u_k having a magnitude p_s. Similar to the Kuhn segment, each solvent molecule is assigned a smeared density distribution of width a_s. In this work, we focus on the effects of permanent dipole moments and do not allow variations in the magnitude of the electric dipole moments. However, generalization to the case of polarizable molecules^50–53 is straightforward.⁴⁹

For studying the effects of added salt, we considered finite sized non-polarizable ions, treated using the smearing function approach.^44,45,48,49 We assume that there are n₊ positive ions (cations) and n₋ negative ions (anions) with valency z₊ and z₋, respectively, so that global electroneutrality is satisfied, i.e., n₊z₊ + n₋z₋ = 0. All of the four components (monomers, solvents, cations, and anions) are assumed to satisfy the incompressibility condition, i.e., the total volume (V) can be expressed as V = n_pN/ρ_p,0 + n_s/ρ_s,0 + n₊/ρ_+,0 + n₋/ρ_−,0, where ρ_p,0, ρ_s,0, ρ_+,0, and ρ_−,0 are the “bulk” monomer, solvent, positive ion, and negative ion number densities, respectively. We should point out that by “bulk,” we mean the region in space outside the brush, where electrostatic potential and densities are spatially independent.

As in our previous studies,^41,44,45 the theory for a planar dipolar brush in an electrolyte solution is developed in a canonical ensemble (i.e., for a fixed number of molecules). A semi-open polymer brush in equilibrium with the electrolyte solution is studied by fixing the chemical potentials of all the molecules that can be exchanged, mainly, solvent, cations, and anions. The partition function for the brush in the canonical ensemble can be written as

\begin{matrix} Z & = & \frac{Λ^{- 3 n}}{n_{s}! n_{+}! n_{-}!} \int_{r_{α} (0)} \prod_{α = 1}^{n_{p}} D [R_{α}] \int \prod_{β = 1}^{n_{s} + n_{+} + n_{-}} d r_{β} \int \prod_{α = 1}^{n_{p}} \prod_{t_{α} = 0}^{N} d u_{α} (t_{α}) \int \prod_{β = 1}^{n_{s}} d u_{β} \exp [- H_{0} \{R_{α}\}) \\ - (H_{w} {R_{α}, r_{β}} - H_{e} {R_{α}, u_{α}, r_{β}, u_{β}}] \prod_{r} δ [\sum_{j = p, s, \pm} \frac{{\hat{ρ}}_{j} (r)}{ρ_{j 0}} - 1], \end{matrix}

(1)

where n = n_pN + n_s + n₊ + n₋ is the total number of particles and Λ is the de Broglie wavelength. H₀ is the Wiener measure for the flexible polymer chains,⁴⁷ given by

H_{0} \{R_{α}\} = \frac{3}{2 b^{2}} \sum_{α = 1}^{n_{p}} \int_{0}^{N} d t_{α} {(\frac{\partial R_{α} (t_{α})}{\partial t_{α}})}^{2},

(2)

and $H_{w} \{R_{α}, r_{β}\}$ takes into account the energetic contributions coming from short-range repulsive hard-core interactions and the attractive dispersive interactions excluding the permanent dipole-dipole interactions. $H_{w}$ can be expressed using Edwards’s formulation⁴⁷ in a form

H_{w} \{R_{α}, r_{β}\} = \frac{1}{2} \sum_{j = p, s, \pm} \sum_{j^{'} = p, s, \pm} \int d r w_{j j^{'}} {\hat{ρ}}_{j} (r) {\hat{ρ}}_{j^{'}} (r),

(3)

where $w_{{j j}^{'}}$ is the excluded volume parameter describing the strength of interactions between particles of type j and j′. It should be noted that Eq. (3) is written with the assumption that the interaction potentials between different pairs can be approximated by delta functions so that the range of interaction is infinitesimal. Furthermore, ${\hat{ρ}}_{p} (r)$ ⁠, ${\hat{ρ}}_{s} (r)$ ⁠, ${\hat{ρ}}_{+} (r)$ ⁠, and ${\hat{ρ}}_{-} (r)$ represent the microscopic number density of the monomers, solvent, cations, and anions, respectively, at a certain location r. They are defined as

{\hat{ρ}}_{p} (r) = \sum_{α = 1}^{n_{p}} \int_{0}^{N} d t_{α} ĥ_{p} [r - R_{α} (t_{α})],

(4)

{\hat{ρ}}_{j = s, \pm} (r) = \sum_{β = 1}^{n_{j}} ĥ_{j} [r - r_{β}],

(5)

ĥ_{j = p, s, \pm} (r) = {[\frac{1}{2 a_{j}^{2}}]}^{3 / 2} \exp [- \frac{π | r |^{2}}{2 a_{j}^{2}}] .

(6)

$ĥ_{j} [r - r^{'}]$ represents the finite distribution of the density of j as a function of distance |r − r′| from the center of mass located at r′. Particular choices of Gaussian functions for $ĥ_{j}$ in Eq. (6) are motivated by physically relevant radially symmetric density distributions and mathematical convenience in handling these functions numerically. In Eq. (1), H_e is the electrostatic contribution resulting from monopole-monopole, monopole-dipole, and dipole-dipole interactions. A simplified expression combining the three types of interaction terms can be written as⁴¹

H_{e} \{R_{α}, u_{α}, r_{β}, u_{β}\} = \frac{l_{B o}}{2} \int d r \int d r^{'} \frac{[{\hat{ρ}}_{e} (r) - \nabla_{r} . {\hat{P}}_{a v e} (r)] [{\hat{ρ}}_{e} (r^{'}) - \nabla_{r^{'}} . {\hat{P}}_{a v e} (r^{'})]}{| r - r^{'} |},

(7)

where l_Bo = e²/4πϵ_ok_BT is the Bjerrum length in vacuum, e being the charge of an electron, ϵ_o is the permittivity of vacuum, k_B is the Boltzmann constant, and T is the temperature (in Kelvin). Moreover, ${\hat{ρ}}_{e} (r) = z_{+} {\hat{ρ}}_{+} (r) + z_{-} {\hat{ρ}}_{-} (r)$ is the local charge density (in units of e). Also, ${\hat{P}}_{a v e} (r^{'}) = \int d u \hat{P} (r, u) u$ ⁠, where $\hat{P} (r, u) = p_{p} {\bar{ρ}}_{p} (r, u) + p_{s} {\bar{ρ}}_{s} (r, u)$ is the local polarization density at r in the direction specified by a unit vector on the surface of a sphere, given by u. Formally, ${\bar{ρ}}_{p} (r, u)$ and ${\bar{ρ}}_{s} (r, u)$ are microscopic number densities of monomeric and solvent dipoles, respectively, and defined by

{\bar{ρ}}_{p} (r, u) = \sum_{α = 1}^{n_{p}} \int_{0}^{N} d t_{α} ĥ_{p} [r - R_{α} (t_{α})] δ [u - u_{α} (t_{α})],

(8)

{\bar{ρ}}_{s} (r, u) = \sum_{β = 1}^{n_{s}} ĥ_{s} [r - r_{β}] δ [u - u_{β}] .

(9)

Using field theoretical transformations,^44–47 we can write the partition function for the brush as

\begin{matrix} Z & = & Z_{o} \int D [ϕ_{p}] \int D [ϕ_{s}] \int D [w_{p}] \\ \times \int D [w_{s}] \int D [η] \exp [- \frac{H_{n e u} + H_{e l e c}}{k_{B} T}], \end{matrix}

(10)

where $Z_{o} = \exp [- F_{0} / k_{B} T]$ is the partition function in the absence of inter-particle interactions, given by

\begin{matrix} \frac{F_{0}}{k_{B} T} & = & \frac{1}{2} [w_{p p} n_{p} N ρ_{p 0} + w_{s s} n_{s} ρ_{s 0}] \\ + \ln [\frac{Λ^{3 n} n_{s}! n_{+}! n_{-}!}{{(4 π)}^{n_{p} N + n_{s}} V^{n_{s} + n_{+} + n_{-}} \prod_{α = 1}^{n_{p}} Q_{p, α} \{0\}}], \end{matrix}

(11)

and we have defined

Q_{p, α} \{0\} = \int_{r_{α} (0)} D [R_{α}] \exp [- \frac{3}{2 b^{2}} \int_{0}^{N} d t_{α} {(\frac{\partial R_{α}}{\partial t_{α}})}^{2}]

(12)

as the partition function for a single Gaussian chain anchored at r_α(0). In Eq. (10), $w_{p}$ and $w_{s}$ are the collective field conjugate to the density variables for monomer (ϕ_p) and solvent (ϕ_s), respectively. η is the pressure field introduced to impose the local incompressibility condition represented by the delta function in Eq. (1). Furthermore,

\begin{matrix} \frac{H_{n e u}}{k_{B} T} & = & {\bar{χ}}_{p s} \int d r \frac{ρ_{p} (r)}{ρ_{p, 0}} \frac{ρ_{s} (r)}{ρ_{s, 0}} - i \sum_{j = p, s} \int d r w_{j} (r) ϕ_{j} (r) \\ + i \int d r η (r) [\sum_{j = p, s} \frac{ρ_{j} (r)}{ρ_{j 0}} - 1] - n_{s} \ln {\bar{Q}}_{s} \{w_{s}\} \\ - \sum_{α = 1}^{n_{p}} \ln {\bar{Q}}_{p, α} \{w_{p}\} . \end{matrix}

(13)

Here, we have defined ρ_j(r) = ∫dr′ $ĥ_{j}$ (r − r′)ϕ_j(r′) along with ${\bar{χ}}_{p s}$ (having a dimension of length⁻³), written as

{\bar{χ}}_{p s} = w_{p s} ρ_{p, 0} ρ_{s, 0} - \frac{w_{p p} ρ_{p, 0}^{2} + w_{s s} ρ_{s, 0}^{2}}{2} .

(14)

In writing Eq. (13), we have assumed that the enthalpy of mixing of all the components except for monomer-solvent pairs is negligible, i.e., (⁠ $w_{i i}$ ρ_i,0 + $w_{j j}$ ρ_j,0)/2 = $w_{i j}$ ρ_i,0ρ_j,0. Moreover, ${\bar{Q}}_{p, α} \{w_{p}\}$ is the normalized partition function for a polymer chain grafted at point r_α(0), given by

{\bar{Q}}_{p, α} \{w_{p}\} = \frac{\int_{r_{α} (0)} D [R_{α}] \exp [- \frac{3}{2 b^{2}} \int_{0}^{N} d t_{α} {(\frac{\partial R_{α}}{\partial t_{α}})}^{2} - i \int_{0}^{N} d t_{α} w_{p} (R_{α})]}{\int_{r_{α} (0)} D [R_{α}] \exp [- \frac{3}{2 b^{2}} \int_{0}^{N} d t_{α} {(\frac{\partial R_{α}}{\partial t_{α}})}^{2}]} .

(15)

Similarly, ${\bar{Q}}_{s}$ is the partition function for a single solvent molecule, given by

{\bar{Q}}_{s} \{w_{s}\} = \frac{1}{V} \int d r \exp [- i w_{s} (r)] .

(16)

The terms in the field theoretic Hamiltonian [cf. Eq. (10)], which explicitly depend on a collective variable (ψ) conjugate to the local charge density (⁠ ${\hat{ρ}}_{e}$ ⁠) and polarization (⁠ ${\hat{P}}_{a v e}$ ⁠), are clubbed together in H_elec so that

\begin{matrix} \exp [- \frac{H_{e l e c}}{k_{B} T}] & = & \frac{1}{ζ_{ψ}} \int D [ψ] \exp [\frac{1}{8 π l_{B o}} \int d r ψ (r) \nabla_{r}^{2} ψ (r)) \\ (+ \sum_{k = \pm} n_{k} \ln {\bar{Q}}_{k} \{ψ, η\}) \\ (+ \sum_{j = p, s} \int d r ϕ_{j} (r) \ln γ_{j} \{ψ, r\}] . \end{matrix}

(17)

Here, we have defined^44,45

γ_{j} \{ψ, r\} = \frac{\sin \{|U_{j} \{ψ, r\}|\}}{|U_{j} \{ψ, r\}|},

(18)

U_{j} \{ψ, r\} = p_{j} \int d r^{'} ψ (r^{'}) \nabla_{r^{'}} ĥ_{j} (r^{'} - r),

(19)

and

ζ_{ψ} = \int D [ψ] \exp [\frac{1}{8 π l_{B o}} \int d r ψ (r) \nabla_{r}^{2} ψ (r)] .

(20)

${\bar{Q}}_{k = \pm}$ is the partition function for a single particle of type k. Explicitly, these are given by

\begin{matrix} {\bar{Q}}_{k = \pm} \{ψ, η\} \\ = \frac{1}{V} \int d r \exp [- i \int d r^{'} ĥ_{k} (r - r^{'}) \{z_{k} ψ (r^{'}) + \frac{η (r^{'})}{ρ_{k, 0}}\}] . \end{matrix}

(21)

Approximating all the functional integrals by invoking the standard saddle-point approximation, a set of coupled non-linear equations can be obtained. However, solving these equations using Dirichlet boundary conditions for the chain propagator reveals that the structure of the dipolar brush is independent of the dipole moments, p_j=p,s, which is unphysical. In order to go beyond the saddle-point approximation, we have approximated the functional integral over ψ using a variational method described below based on Dyson-Schwinger’s approach from quantum field theory.⁵⁴ After approximating the functional integral over ψ, the other functional integrals are evaluated at the saddle-point, which leads to a theory suitable for studying the effects of dipolar interactions and salt ions on the monomer density profiles of planar polymer brushes.

A. Electrostatics using the Dyson-Schwinger equations

Here, we present details of the procedure for approximating the functional integral over ψ [cf. Eq. (17)]. The procedure is based on the method leading to the Dyson-Schwinger equations in quantum field theory. In particular, due to the fact that the integral of a total derivative is zero,⁵⁴ we can write, for any path integral and for our case, ψ,

\int D [ψ] \frac{δ}{δ ψ} \exp [- S \{ψ, ϕ_{j}, η\} + \int d r J (r) ψ (r)] = 0

(22)

with action, S, and J as an arbitrary field variable, which will be replaced by zero in the end. For Eq. (17), the action is a functional of ψ, ϕ_j, and η, given by

\begin{matrix} S \{ψ, ϕ_{j}, η\} & = & - \frac{1}{8 π l_{B o}} \int d r ψ (r) \nabla_{r}^{2} ψ (r) - \sum_{k = \pm} n_{k} \ln {\bar{Q}}_{k} \{ψ, η\} \\ - \sum_{j = p, s} \int d r ϕ_{j} (r) \ln γ_{j} {ψ, r} . \end{matrix}

(23)

Evaluating the functional derivative in Eq. (22) formally with the action $S \{ψ, ϕ_{j}, η\}$ ⁠, and putting J = 0, leads to

{⟨\frac{δ S \{ψ, ϕ_{j}, η\}}{δ ψ (r)}⟩}_{S} = 0,

(24)

where

{⟨ \dots ⟩}_{S} = \frac{\int D [ψ] (\dots) \exp [- S \{ψ, ϕ_{j}, η\}\}}{\int D [ψ] \exp [- S \{ψ, ϕ_{j}, η\}\}} .

(25)

Taking an additional derivative of Eq. (22) with respect to J(r) and putting J = 0 lead to the equation

{⟨\frac{δ S \{ψ, ϕ_{j}, η\}}{δ ψ (r)} ψ (r^{'})⟩}_{S} = δ (r - r^{'}) .

(26)

Although formally exact, the averages defined in Eqs. (24) and (26) cannot be computed easily. As an approximation, we consider a variational/trial function approach and approximate

{⟨ \dots ⟩}_{S} ≃ {⟨ \dots ⟩}_{S_{t}},

(27)

where S_t is a trial function and chosen to be

\begin{matrix} S_{t} \{ψ, ϕ_{j}, η\} & = & \frac{1}{2} \int d r \int d r^{'} [ψ (r) - \bar{ψ} (r)] \\ \times K^{- 1} (r, r^{'}, \{ϕ_{j}, η\}) [ψ (r^{'}) - \bar{ψ} (r^{'})], \end{matrix}

(28)

where $\bar{ψ} (r)$ and $K^{- 1} (r, r^{'}, \{ϕ_{j}, η\})$ are unknowns and need to be determined by solving the approximate Dyson-Schwinger equations, written as [from Eqs. (24) and (26)]

{⟨\frac{δ S \{ψ, ϕ_{j}, η\}}{δ ψ (r)}⟩}_{S_{t}} = 0,

(29)

{⟨\frac{δ S \{ψ, ϕ_{j}, η\}}{δ ψ (r)} ψ (r^{'})⟩}_{S_{t}} = δ (r - r^{'}) .

(30)

Explicitly, for the action given by Eq. (23), Eq. (29) can be written as (see Appendix A for the details of the derivation)

\nabla_{r} . [\int d r_{1} ϵ (r, r_{1}) \nabla_{r_{1}} i \bar{ψ} (r_{1})] = - 4 π l_{B o} ρ_{e} (r),

(31)

where ϵ(r, r₁) is a non-local dielectric function, given by

\begin{matrix} ϵ (r, r_{1}) & = & δ (r - r_{1}) + 4 π l_{B o} \int d r_{2} \sum_{j = p, s} p_{j}^{2} ϕ_{j} (r_{2}) ĥ_{j} (r - r_{2}) \\ \times ĥ_{j} (r_{2} - r_{1}) {\bar{G}}_{j} (| U_{j} {i \bar{ψ}, r_{2}} |), \end{matrix}

(32)

and ρ_e(r) is collective local charge density, given by

ρ_{e} (r) = \sum_{k = \pm} z_{k} ρ_{k} (r),

(33)

ρ_{k} (r) = \int d r_{2} ĥ_{k} (r - r_{2}) ϕ_{k} (r_{2}) .

(34)

Here, we have defined

ϕ_{k = \pm} (r) = \frac{n_{k} \exp [- i \int d r^{'} ĥ_{k} (r - r^{'}) \{z_{k} \bar{ψ} (r^{'}) + \frac{η (r^{'})}{ρ_{k, 0}}\} - E_{k} (r)]}{\int d r^{″} \exp [- i \int d r^{'} ĥ_{c} (r^{″} - r^{'}) \{z_{k} \bar{ψ} (r^{'}) + \frac{η (r^{'})}{ρ_{k, 0}}\} - E_{k} (r^{″})]},

(35)

{\bar{G}}_{j = p, s} (| U_{j} {i \bar{ψ}, r_{2}} |) = \frac{L (| U_{j} {i \bar{ψ}, r_{2}} |)}{| U_{j} {i \bar{ψ}, r_{2}} |},

(36)

where L(x) = coth x − 1/x is the Langevin function. Also,

\begin{matrix} E_{k} (r) = \frac{z_{k}^{2}}{2} \int d r_{1} \int d r_{2} ĥ_{k} (r - r_{1}) K (r_{1}, r_{2}, {ϕ_{j}, η}) ĥ_{k} (r - r_{2}), \end{matrix}

(37)

which needs to be determined by solving Eq. (30), which can be written as (see Appendix B for the derivation)

\begin{matrix} \nabla_{r}^{2} K (r, r^{'}, {ϕ_{j}, η}) - \int d r_{3} τ^{2} (r, r_{3}) K (r_{3}, r^{'}, {ϕ_{j}, η}) \\ + 4 π l_{B o} \int d r_{1} \sum_{j = \pm} ϕ_{j} (r_{1}) \int d r_{3} {[\frac{δ^{2} \ln γ_{j} {ψ, r_{1}}}{δ ψ (r_{3}) δ ψ (r)}]}_{ψ = \bar{ψ}} \\ \times K (r_{3}, r^{'}, {ϕ_{j}, η}) = - 4 π l_{B o} δ (r - r^{'}), \end{matrix}

(38)

where

\begin{matrix} τ^{2} (r, r_{3}) & = & 4 π l_{B o} \int d r_{1} \sum_{k = \pm} z_{k}^{2} ĥ_{k} (r - r_{1}) ϕ_{k} (r_{1}) ĥ_{k} (r_{1} - r_{3}) . \end{matrix}

(39)

An approximation for the functional integral over ψ [cf. Eq. (17)] can be obtained as follows:

\begin{matrix} \exp [- \frac{H_{e l e c}}{k_{B} T}] & = & {⟨\exp [- [S \{ψ, ϕ_{j}, η\} - S_{t} \{ψ, ϕ_{j}, η\}]]⟩}_{S_{t}} \frac{\int D [ψ] \exp [- S_{t} \{ψ, ϕ_{j}, η\}]}{ζ_{ψ}} \\ ≃ & \exp [- {⟨S \{ψ, ϕ_{j}, η\} - S_{t} \{ψ, ϕ_{j}, η\}⟩}_{S_{t}}] \frac{\int D [ψ] \exp [- S_{t} \{ψ, ϕ_{j}, η\}]}{ζ_{ψ}}, \end{matrix}

(40)

= \exp [- \frac{H_{e l e c, D S} \{ϕ_{j}, η\}}{k_{B} T}],

(41)

where we have used ${⟨e^{- x}⟩}_{S_{t}} ≃ 1 - {⟨x⟩}_{S_{t}} ≃ e^{- {⟨x⟩}_{S_{t}}}$ ⁠. Equation (41) can be written in a form (see Appendix B)

\begin{matrix} \frac{H_{e l e c, D S} \{ϕ_{j}, η\}}{k_{B} T} & = & - \frac{1}{8 π l_{B o}} \int d r \bar{ψ} (r) \nabla_{r}^{2} \bar{ψ} (r) + i \int d r ρ_{e} (r) \bar{ψ} (r) + i \int d r η (r) [\sum_{k = \pm} \frac{ρ_{k} (r)}{ρ_{k, 0}}] \\ - \int d r \sum_{j = p, s} ϕ_{j} (r) \ln γ_{j} {\bar{ψ}, r} + \int d r \sum_{k = \pm} ϕ_{k} (r) \ln [ϕ_{k} (r) V / n_{k}] \\ - \ln [\frac{1}{ζ_{ψ}} \int D [ψ] e^{- S_{t} \{ψ, ϕ_{j}, η\}}], \end{matrix}

(42)

where $\bar{ψ}$ and K appearing in ρ_e and the last term need to be computed using Eqs. (31) and (38), respectively.

In Sec. III, we present an application of the field theory developed here to study a planar polymer brush in equilibrium with an electrolyte solution containing symmetric ions.

III. PLANAR POLYMER BRUSH IN EQUILIBRIUM WITH A SALTY SOLUTION

A. Symmetric salt ions: Local electroneutrality

Here, we consider a planar polymer brush immersed in an electrolyte solution containing symmetric salt ions so that the ions are identical in their sizes and magnitude of charges. For the symmetric salt ions, we use the notation $ĥ_{k = \pm} = ĥ_{s a l t}$ ⁠, |z_k=±| = z_salt, n_k=± = n_salt, ρ_k,0 = ρ_salt,0, ϕ_k=± = ϕ_salt, ρ_k=± = ρ_salt, and E_k=± = E_salt. Furthermore, by symmetry, local electroneutrality will hold so that ρ_e(r) = 0, which leads to $i \bar{ψ} (r) = ψ_{b}$ as the solution of Eq. (31), where ψ_b is the electrostatic potential in the solution far from the brush. Furthermore, this leads to

\begin{matrix} {[\frac{δ^{2} \ln γ_{j} {ψ, r_{1}}}{δ ψ (r_{3}) δ ψ (r)}]}_{ψ = \bar{ψ}} = - \frac{p_{j}^{2}}{3} \nabla_{r_{3}} ĥ_{j} (r_{3} - r_{1}) \cdot \nabla_{r} ĥ_{j} (r_{1} - r) \end{matrix}

(43)

so that Eq. (38) becomes

\begin{matrix} \nabla_{r} . [\int d r_{3} ϵ_{W C L} (r, r_{3}) \nabla_{r_{3}} K (r_{3}, r^{'}, {ϕ_{j}, η})] \\ - \int d r_{3} τ^{2} (r, r_{3}) K (r_{3}, r^{'}, {ϕ_{j}, η}) = - 4 π l_{B o} δ (r - r^{'}), \end{matrix}

(44)

where ϵ_WCL(r, r₃) is a non-local dielectric function, given by

\begin{matrix} ϵ_{W C L} (r, r_{3}) & = & δ (r - r_{3}) + \frac{4 π l_{B o}}{3} \int d r_{2} \\ \times \sum_{j = p, s} p_{j}^{2} ϕ_{j} (r_{2}) ĥ_{j} (r - r_{2}) ĥ_{j} (r_{2} - r_{3}), \end{matrix}

(45)

and corresponds to the weak coupling limit⁴⁴ (WCL) of Eq. (32) (i.e., |U_j| → 0). In this work, we approximate ϵ_WCL(r, r₃) ≃ ϵ_l(r)δ(r − r₃), where

ϵ_{l} (r) = 1 + \frac{4 π l_{B o}}{3} \int d r_{2} \sum_{j = p, s} p_{j}^{2} ϕ_{j} (r_{2}) ĥ_{j} (r - r_{2}),

(46)

= 1 + \frac{4 π l_{B o}}{3} \sum_{j = p, s} p_{j}^{2} ρ_{j} (r),

(47)

which is the dominant contribution to the dielectric function in the limit of a_j → 0. Furthermore, ignoring non-local effects resulting from gradients of the local dielectric function (ϵ_l(r)) and effects arising from τ²(r, r₃),

K (r, r^{'}, {ϕ_{j}, η}) = \frac{l_{B o}}{| r - r^{'} | \sqrt{ϵ_{l} (r) ϵ_{l} (r^{'})}} .

(48)

See Ref. 44 for the details of the derivation. It should be noted here that the right-hand side in Eq. (48) has no explicit dependence on η due to the fact that we have neglected effects of salt ions on K in this work. Using these simplifications, Eq. (42) can be transformed into

\begin{matrix} \frac{H_{e l e c, D S} \{ϕ_{j}, η\}}{k_{B} T} & = & \frac{1}{2} \int d r \frac{\ln ϵ_{l} (r)}{ϵ_{l} (r) - 1} [f_{l} (r) + f_{s a l t} (r)] \\ - 2 \int d r ϕ_{s a l t} (r) E_{s a l t} (r) \\ - 2 n_{s a l t} \ln {\hat{Q}}_{s a l t} {ϕ_{j}, η}, \end{matrix}

(49)

where

\begin{matrix} f_{l} (r) & = & \frac{l_{B o}}{3} \int d r^{'} \sum_{j = p, s} p_{j}^{2} ϕ_{j} (r^{'}) \nabla_{r} ĥ_{j} (r - r^{'}) \\ \cdot \int d r^{″} \frac{\nabla_{r^{″}} ĥ_{j} (r^{″} - r^{'})}{| r - r^{″} |}, \end{matrix}

(50)

\begin{matrix} f_{s a l t} (r) & = & 2 l_{B o} z_{s a l t}^{2} \int d r^{'} ϕ_{s a l t} (r^{'}) ĥ_{s a l t} (r - r^{'}) \\ \times \int d r^{″} \frac{ĥ_{s a l t} (r^{″} - r^{'})}{| r - r^{″} |}, \end{matrix}

(51)

\begin{matrix} E_{s a l t} (r) & = & \frac{l_{B o} z_{s a l t}^{2}}{2} \int d r^{'} \frac{ĥ_{s a l t} (r - r^{'})}{ϵ_{l} (r^{'})} \int d r^{″} \frac{ĥ_{s a l t} (r^{″} - r^{'})}{| r - r^{″} |}, \end{matrix}

(52)

\begin{matrix} {\hat{Q}}_{s a l t} {ϕ_{j}, η} \\ = \frac{1}{V} \int d r \exp [- E_{s a l t} (r) - i \int d r^{'} ĥ_{s a l t} (r - r^{'}) \frac{η (r^{'})}{ρ_{s a l t, 0}}], \end{matrix}

(53)

\begin{matrix} ϕ_{s a l t} (r) \\ = \frac{n_{s a l t}}{V {\hat{Q}}_{ϕ_{j}, s a l t} {η}} \exp [- E_{s a l t} (r) - i \int d r^{'} ĥ_{s a l t} (r - r^{'}) \frac{η (r^{'})}{ρ_{s a l t, 0}}], \end{matrix}

(54)

and we have suppressed the functional dependencies of ϵ_l, f_l, f_salt, ϕ_salt, and E_salt on ϕ_j for convenience in writing.

B. Saddle-point approximation and equilibrium with the electrolyte solution

An inhomogeneous polymer brush in equilibrium with a salty solution bath can be studied using the saddle-point approximation⁴⁶ in Eq. (10). The approximation evaluates the functional integrals over the fields by the value of the integrand at the saddle-point. At the saddle-point, the free energy is given by

\frac{F}{k_{B} T} = - \ln Z = \frac{F_{0}}{k_{B} T} + \frac{H_{n e u}}{k_{B} T} + \frac{H_{e l e c, D S}}{k_{B} T},

(55)

where the right-hand side is evaluated at the saddle-point. For a dipolar brush in equilibrium with a salty solution bath, one has to equate the chemical potentials of all the components that can be exchanged between the bulk solution and the brush region. Also, at equilibrium, the osmotic pressure must be the same everywhere. However, due to the incompressibility constraint, equating the chemical potentials of the salt ions and solvent molecules is sufficient to define the equilibrium state of the system. Furthermore, it is assumed that the densities of different components are known in the bulk solution and we use them as parameters in the study here.

Using the thermodynamic relation between the free energy and chemical potentials in the canonical ensemble (i.e., $μ_{k} = {(\partial F / \partial n_{k})}_{V, T}$ ⁠), we can write Eq. (55) as F = F_ref + ΔF, where F_ref = μ_sn_s + μ₊n₊ + μ₋n₋ ≡ μ_sn_s + μ_saltn_salt, i.e., we take the electrolyte solution without the brush as a reference. Explicitly, ΔF is given by

\begin{matrix} \frac{Δ F}{k_{B} T} & = & {\bar{χ}}_{p s} \int d r \frac{ρ_{p} (r)}{ρ_{p, 0}} \frac{ρ_{s}^{g c} (r)}{ρ_{s, 0}} - i \int d r [w_{p} (r) ϕ_{p} (r) + w_{s} (r) ϕ_{s}^{g c} (r)] \\ + i \int d r η (r) [\frac{ρ_{p} (r)}{ρ_{p, 0}} + \frac{ρ_{s}^{g c} (r)}{ρ_{s, 0}} - 1] - \sum_{α = 1}^{n_{p}} \ln {\bar{Q}}_{p, α} \{w_{p}\} \\ + \frac{1}{2} \int d r \frac{\ln ϵ_{l}^{g c} (r)}{ϵ_{l}^{g c} (r) - 1} f_{l}^{g c} (r) - \int d r [ϕ_{s}^{g c} (r) + 2 ϕ_{s a l t}^{g c} (r)], \end{matrix}

(56)

where superscript “gc” implies the grand-canonical analog of the density profiles, which highlights exchange of molecules between the bulk solution and the brush. Specifically, the saddle point approximations for the Hamiltonian appearing in Eq. (10) and fixing the chemical potentials of the solvent as well as the salt ions leads to

ϕ_{s}^{g c} (r) = ρ_{s, 0} \exp [- i \{w_{s} (r) - w_{s}^{b}\}],

(57)

\begin{matrix} ϕ_{s a l t}^{g c} (r) & = & ρ_{s a l t, 0} \exp [) - \{E_{s a l t} (r) - E_{s a l t}^{b}\} \\ - i \int d r^{'} ĥ_{s a l t} (r - r^{'}) \frac{η (r^{'}) - η^{b}}{ρ_{s a l t, 0}}]), \end{matrix}

(58)

and $ρ_{s}^{g c} (r) = \int d r^{'} ĥ_{s a l t} (r - r^{'}) ϕ_{s}^{g c} (r^{'})$ ⁠. Here, quantities with superscript b represent values in the bulk. We should point out that in getting Eq. (58), we have ignored a spatially invariant term, which depends on the integrals of ϵ_l(r). The saddle-point approximation with respect to $w_{p}$ gives

ϕ_{p} (r) = \sum_{α = 1}^{n_{p}} \frac{\int_{0}^{N} d t_{α} q_{r_{α}} (r, t_{α}) \bar{q} (r, N - t_{α})}{\int d r^{'} q_{r_{α}} (r^{'}, t_{α}) \bar{q} (r^{'}, N - t_{α})},

(59)

where $\bar{q} (r, N - t_{α})$ satisfies

\frac{\partial \bar{q} (r, t_{α}^{'})}{\partial t_{α}^{'}} = [\frac{b^{2}}{6} \nabla_{r}^{2} - i w_{p} (r)] \bar{q} (r, t_{α}^{'}),

(60)

with the condition $\bar{q} (r, 0) = 1$ for $t_{α}^{'} = N - t_{α} = 0$ ⁠. Similarly, $q_{r_{α}} (r, t_{α})$ satisfies the same equation but with the initial condition $q_{r_{α}} (r, 0) = δ (r - r_{α})$ ⁠. Furthermore, the fields, $w_{s}$ (r) and $w_{p}$ (r), can be approximated by

i w_{s} (r) = \int d r^{'} ĥ_{s} (r - r^{'}) [{\bar{χ}}_{p s} \frac{ρ_{p} (r^{'})}{ρ_{p, 0} ρ_{s, 0}} + \frac{i η (r^{'})}{ρ_{s, 0}} + \frac{2 π l_{B o} p_{s}^{2}}{3} \frac{f_{l}^{g c} (r^{'})}{ϵ_{l}^{g c} (r^{'}) (ϵ_{l}^{g c} (r^{'}) - 1)}] - 2 \int d r^{'} \frac{δ ϕ_{s a l t}^{g c} (r^{'})}{δ ϕ_{s} (r)}

(61)

and

i w_{p} (r) = \int d r^{'} ĥ_{p} (r - r^{'}) [{\bar{χ}}_{p s} \frac{ρ_{s}^{g c} (r^{'})}{ρ_{p, 0} ρ_{s, 0}} + \frac{i η (r^{'})}{ρ_{p, 0}} + \frac{2 π l_{B o} p_{p}^{2}}{3} \frac{f_{l}^{g c} (r^{'})}{ϵ_{l}^{g c} (r^{'}) (ϵ_{l}^{g c} (r^{'}) - 1)}] - 2 \int d r^{'} \frac{δ ϕ_{s a l t}^{g c} (r^{'})}{δ ϕ_{p} (r)} .

(62)

Finally, optimization with respect to the pressure field η gives

\frac{ρ_{p} (r)}{ρ_{p, 0}} + \frac{ρ_{s}^{g c} (r)}{ρ_{s, 0}} + 2 \frac{ρ_{s a l t}^{g c} (r)}{ρ_{s a l t, 0}} = 1 .

(63)

C. Laterally homogeneous brush: One-dimensional model

We solved Eqs. (57)–(63) assuming lateral homogeneity in a planar dipolar brush. Furthermore, molar volumes of a polymer segment and solvent were assumed to be the same, i.e., ρ_p,0 = ρ_s,0 = ρ₀ = 1/b³. Numerical solution of the equations gets considerably easier in the limit when polymer segments, salt ions, and solvent molecules have small sizes. In particular, in the limits of a_j → 0, these equations get simplified and we can circumvent the problem of numerically evaluating convolutions over $ĥ_{j}$ ⁠. The simplification results from the identity $ĥ_{j}$ (r) → δ(r) for a_j → 0. However, we retained distinguishability between the solvent as well as segment sizes (a_p = a_s = a) and sizes of the salt ions, a_salt.

These simplifications lead to [cf. Eqs. (61) and (62)]

\begin{matrix} i w_{s} (x) & = & {\bar{χ}}_{p s} \frac{ρ_{p} (x)}{ρ_{0}^{2}} + \frac{i η (x)}{ρ_{0}} + \frac{1}{16 a^{3} ρ_{0}} \frac{ϵ_{s} - 1}{ϵ (x)} - \frac{ϵ_{s}}{ϵ^{2} (x) ρ_{0}} \\ \times ρ_{s a l t}^{g c} (x) [\frac{z_{s a l t}^{2} l_{B o}}{a_{s a l t}}] \end{matrix}

(64)

and

\begin{matrix} i w_{p} (x) & = & {\bar{χ}}_{p s} \frac{ρ_{s}^{g c} (x)}{ρ_{0}^{2}} + \frac{i η (x)}{ρ_{0}} + \frac{1}{16 a^{3} ρ_{0}} \frac{ϵ_{p} - 1}{ϵ (x)} - \frac{ϵ_{p}}{ϵ^{2} (x) ρ_{0}} \\ \times ρ_{s a l t}^{g c} (x) [\frac{z_{s a l t}^{2} l_{B o}}{a_{s a l t}}], \end{matrix}

(65)

where the local dielectric function of the medium (ϵ(x) ≡ ϵ_l(x)) is given by [cf. Eq. (47)]

ϵ (x) = 1 + (ϵ_{p} - 1) \frac{ρ_{p} (x)}{ρ_{0}} + (ϵ_{s} - 1) \frac{ρ_{s} (x)}{ρ_{0}} .

(66)

Here, we have defined ϵ_p and ϵ_s as the dielectric constants of the homogeneous polymer and solvent, respectively. Explicitly, these are defined as

ϵ_{j = p, s} = 1 + \frac{4 π l_{B o} ρ_{0}}{3} p j^{2} .

(67)

$ρ_{s}^{g c} (x) \equiv ϕ_{s}^{g c} (x)$ and $ρ_{s a l t}^{g c} (x) \equiv ϕ_{s a l t}^{g c} (x)$ are the number densities of solvent and salt ions, respectively, at x [cf. Eqs. (57) and (58)], given by

ρ_{s}^{g c} (x) = ρ_{0} \exp [- i \{w_{s} (x) - w_{s}^{b}\}],

(68)

\begin{matrix} ρ_{s a l t}^{g c} (x) & = & ρ_{s a l t, 0} \exp [- i \frac{(η (x) - η^{b})}{ρ_{s a l t, 0}}] \\ \times \exp [- \frac{z_{s a l t}^{2}}{2 a_{s a l t}} \{\frac{1}{ϵ (x)} - \frac{1}{ϵ_{b}}\}] . \end{matrix}

(69)

Here, quantities with superscript b represent the bulk homogeneous system. Furthermore, the local incompressibility condition [cf. Eq. (63)] becomes

\frac{ρ_{p} (x)}{ρ_{0}} + \frac{ρ_{s}^{g c} (x)}{ρ_{0}} + 2 \frac{ρ_{s a l t}^{g c} (x)}{ρ_{s a l t, 0}} = 1 .

(70)

Similarly, Eq. (59) gives

ρ_{p} (x) \equiv ϕ_{p} (x) = \sum_{α = 1}^{n_{p}} \frac{\int_{0}^{N} d t_{α} q_{x_{α}} (x, t_{α}) \bar{q} (x, N - t_{α})}{\int d x^{'} q_{x_{α}} (x^{'}, t_{α}) \bar{q} (x^{'}, N - t_{α})},

(71)

where $\bar{q} (x, N - t_{α})$ satisfies

\frac{\partial \bar{q} (x, t_{α}^{'})}{\partial t_{α}^{'}} = [\frac{b^{2}}{6} \nabla_{x}^{2} - i w_{p} (x)] \bar{q} (x, t_{α}^{'})

(72)

with the condition $\bar{q} (x, 0) = 1$ for $t_{α}^{'} = N - t_{α} = 0$ ⁠. Similarly, $q_{x_{α}} (x, t_{α})$ satisfies the same equation but with the initial condition $q_{x_{α}} (x, 0) = δ (x - x_{α})$ ⁠.

Equations (64)–(72) are solved self-consistently for obtaining seven different spatial variables: $i η (x), i w_{s} (x), i w_{p} (x), ϵ (x), ρ_{s}^{g c} (x), ρ_{s a l t}^{g c} (x)$ ⁠, and ρ_p(x) until the free energy of the system was invariant with a tolerance of 10⁻⁸. All of the quantities having dimensions of length were made dimensionless by dividing them by $R_{g o} = {(N b^{2} / 6)}^{1 / 2}$ ⁠. A box length of 15R_go was required for ensuring that the fields far from the brush region approached zero. Numerical results presented in this work were obtained by taking the maximum box length of 25R_go with 256 grid points. The modified diffusion equation represented by Eq. (72) was solved by using an implicit-explicit scheme known as the extrapolated gear method.^55,56 Dirichlet boundary conditions were used for q_xα and $\bar{q}$ at the substrate, i.e., $q_{x α} (x = 0, t_{α}) = \bar{q} (x = 0, t_{α}) = 0$ for all values of t_α. The grafted ends were displaced to the first grid point in the solution so that δ = 0.098R_go for a box length of 25R_go with 256 grid points. Also, a chain contour step of 10⁻⁴ was used for the time stepping while solving the modified diffusion equations. We started from an initial guess for the fields (⁠ $i w_{p}$ ⁠, $i w_{s}$ ⁠) and computed the values of the fields by solving the modified diffusion equation while updating iη(x) using the set of non-linear equations described above. The guessed and the computed values for $i w_{p}$ ⁠, ${i w}_{s}$ were mixed using the simple mixing scheme⁴⁶ to develop a new guess for the next iteration. The iterative procedure was continued until the free energy of the brush (with respect to the solution bath) was invariant with a tolerance of 10⁻⁸.

IV. RESULTS

In the absence of salt ions, effects of dipolar interactions on a planar polymer brush have already been described in our previous work.⁴⁴ It has been shown that $Δ ϵ = 4 π l_{B o} (p_{p}^{2} - p_{s}^{2}) ρ_{0} / 3 ϵ_{s} = (ϵ_{p} / ϵ_{s} - 1)$ is a key parameter. The dipolar interactions lead to the renormalization of the polymer-solvent interaction parameter (⁠ ${\bar{χ}}_{p s, e f f}$ ⁠) so that we can write

{\bar{χ}}_{p s, e f f} - {\bar{χ}}_{p s} \sim Δ ϵ^{2} / a^{3} .

(73)

Equation (73) reveals that an effective quality of the solvent decreases with an increase in the dipolar mismatch (i.e., Δϵ) irrespective of its sign. In other words, a planar polymer brush always shrinks due to an increase in the dipolar mismatch between the monomer and the solvent and the extent of shrinkage of the polymer brush depends only on the magnitude of the mismatch.

In the following, we present the effects of added symmetric salt ions on the polymer segment density profiles. Parameters for the salt ions were chosen so that the cation sizes were set equal to the anion sizes. Four different anions were selected: Cl⁻, I⁻, and two more ion types twice the size of the Cl⁻ and I⁻. Parameters for the ions studied in this work are listed in Table I. The spatial variation of the uncharged additives and the salt ions is also presented in the units of M(mol/l) along with their volume fraction. The concentration in mol/l $= ϕ_{+} (x) ρ_{s a l t, 0} / 0.6023 = ϕ_{-} (x) ρ_{s a l t, 0} / 0.6023$ ⁠, where the factor of 0.6023 in the denominator is for conversion from molecules/nm³ to mol/l, and the numerical values for ρ_salt,0 for the salts/additives are listed in Table I.

TABLE I.

Parameters for the salt ions studied in this work.

Ion	a_salt (nm)	1/ρ_salt,0 (nm³/molecule)
Cl⁻	0.12	0.0496
I⁻	0.15	0.0734
2Cl⁻	0.24	0.0992
2I⁻	0.30	0.1468

Parameters for Cl⁻ and I⁻ were obtained from Ref. 57. The parameter a_salt was normalized by dividing with a factor of π^1/22^−1/3 to be consistent with the Gaussian definition in Ref. 57, and the molar volume of the ions was converted from cm³/mol to nm³/molecule. In order to decipher the effect of crowding due to finite sized ions resulting from the local incompressibility constraint, translational entropy, and ion-solvation, we first present the results for a case where we deliberately switch off the charge on the ions (i.e., z_k = 0). In this case, effects resulting from translational entropy and local incompressibility constraint persist without having any contribution from ion-solvation effects [i.e., the last terms on the right-hand side in Eqs. (61) and (62)].

Other parameters used in describing the polymer-solvent system were estimated to be b = 1 nm such that ρ₀ = 1 nm⁻³, a = 0.34 nm, ${\bar{χ}}_{p s} / ρ_{0} =$ 0.1, σ = 0.075 chains nm⁻², ϵ_p = 60, and ϵ_s = 80. The choice for b was motivated by the fact that for the most polymers in a good solvent, the Kuhn segment is of the order of nm. ${\bar{χ}}_{p s} / ρ_{0} = 0.1$ was chosen to describe a good solvent-like condition. The dielectric constant of the pure solvent (ϵ_s) was set at 80 to represent water at room temperature, and ϵ_p = 60 represents a highly polar polymer. Dielectric constants of different polymers are reported in the Polymer Handbook.⁵⁸ The value for the grafting density was chosen to avoid laterally inhomogeneous brushes and is based on our previous work.⁴⁴ Similarly, the value of a was chosen so that the ratio a/b < 1. We should point out that $a, {\bar{χ}}_{p s} / ρ_{0}$ and (ϵ_p − ϵ_s)/ϵ_s can be obtained by fitting experimental data for polymer concentration dependent osmotic pressure, as described in our previous work.⁴⁴ Also, temperature (T) dependence in the theoretical framework presented here appear via ${\bar{χ}}_{p s}$ (typically, assumed to be of the form, A + B/T) and l_Bo ∼ 1/T.

A. Uncharged additives

Purely entropic effects of the ions were investigated by setting the charge of the ions to zero while solving the equations presented above. It should be noted that temperature dependent excluded volume interactions are not present in this case, and setting the charge of the ions to zero lets us explore purely entropic effects resulting from the translational entropy and local incompressibility constraint (or crowding due to finite sized additives). As shown in Fig. 2(a), the polymer brush expanded irrespective of the size of the uncharged additives. The expansion resulted from the penetration of the additives in the interior of the brush [Fig. 2(b)], which was found to be almost uniform. The uniform distribution of additives was expected due to the fact that translational entropy gets maximized if the additives explore the whole space. However, an increase in the size of the additives led to slightly lower concentration in the interior of the brush in comparison with their bulk value, which reveal that the crowding effects hinder uniform distribution to some extent. For example, the volume fraction of Cl⁻ in the bulk and the interior was almost equal while the volume fraction of 2I⁻ in the interior was slightly lower than its bulk value in Fig. 2(b). Furthermore, the volume fraction of the solvent was reduced in the bulk due to the presence of the additives and almost the same reduction in the solvent volume fraction was seen in the interior of the brush. Such “dehydration” of PNIPAM brushes due to the addition of salt ions has been reported by Matsuoka and Uda³⁵, and we obtain qualitatively similar results due to purely entropic effects considered here. It should be noted that the expansion of the brush resulted from the incompressibility constraint used in the absence of any interactions with the additives. In order to capture the possible collapse of chains due to the uncharged additives, we have to consider short-ranged interactions with them and effects of concentration fluctuations, both of which can cause effective attraction between polymer segments.^46,47 In summary, entropic effects always lead to expansion of the brush and favor uniform distribution of the additives in the whole space.

FIG. 2.

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Entropic effects of uncharged finite-sized additives on the spatial distribution of (a) monomers, (b) additives, and (c) solvent. A bulk additive concentration of 0.5M was used, and all the other parameters are described in the main text. The polymer brush exhibited expansion [panel (a)] for all the cases Cl⁻, I⁻, 2Cl⁻, and 2I⁻ relative to the additive-free solution due to penetration of the additives in the interior of the brush [panel (b)]. In the absence of any interactions with the additives, the presence of the additives in the interior is purely entropy driven.

B. Effects of ion-solvation, translational entropy, and crowding

For very small ions so that 1/ρ_salt,0 → 0, one can ignore contributions from the ions to Eq. (63) and the ionic distribution [cf. Eq. (58)] becomes independent of the pressure field, η. These simplifications lead to the relation

\begin{matrix} i w_{p} (x) - i w_{s} (x) & = & \frac{{\bar{χ}}_{p s}}{ρ_{0}} [1 - 2 \frac{ϕ_{p} (x)}{ρ_{0}}] + \frac{Δ ϵ}{ϵ (x)} \frac{ϵ_{s}}{16 ρ_{0} a^{3}} \\ - \frac{κ^{2}}{4 π a_{s a l t} ρ_{0}} \frac{Δ ϵ}{ϵ^{2} (x)} \\ \times \exp [- \frac{l_{B 0} z_{s a l t}^{2}}{2 a_{s a l t}} \{\frac{1}{ϵ (x)} - \frac{1}{ϵ_{s}}\}], \end{matrix}

(74)

where ϵ(x) = ϵ_s(1 + Δϵϕ_p(x)/ρ₀) and $κ^{2} = 8 π l_{B o} z_{s a l t}^{2} ρ_{s a l t, 0}$ ⁠. $i w_{p}$ (x) − ${i w}_{s}$ (x) is the exchange potential for replacing a solvent by a monomer. In the limit of Δϵϕ_p(x) → 0, an effective polymer-solvent interaction parameter can be written as

{\bar{χ}}_{p s, e f f} = {\bar{χ}}_{p s} + \frac{Δ ϵ^{2}}{32 a^{3}} - \frac{κ^{2} Δ ϵ^{2}}{16 π a_{s a l t}} [4 - \frac{l_{B 0} z_{s a l t}^{2}}{a_{s a l t} ϵ_{s}}]

(75)

which is the prefactor for the linear term in the expansion representing $i w_{p}$ (x) in terms of ϕ_p(x). Equation (75) reveals that for a given monomer-solvent pair, the polymer brush can either contract or expand depending on the size of the salt ions. Noting that l_B0/ϵ_s is the Bjerrum length in the solvent, the Bjerrum length and the size of the salt ion dictate the structure of the dipolar polymer brush. In particular, if $l_{B 0} z_{s a l t}^{2} / (a_{s a l t} ϵ_{s}) > 4$ ⁠, then the polymer is expected to shrink; otherwise the polymer brush is expected to expand. $l_{B 0} z_{s a l t}^{2} / (a_{s a l t} ϵ_{s})$ for the four different salt cases considered here were 5.83, 4.99, 2.91, and 2.50 for Cl⁻, I⁻, 2Cl⁻, and 2I⁻, respectively. Quantitatively a similar dependence of an effective pair-wise interaction parameter on salt concentration and size was derived by Wang⁵⁹ for binary polymer blends of different dielectric constants without the $\frac{Δ ϵ^{2}}{32 a^{3}}$ term. In their analysis, $l_{B 0} z_{s a l t}^{2} / a_{s a l t} ϵ$ is the key parameter that determines the degree of miscibility of the blends due to the addition of salt ions while using a Born model for ion-solvation in a uniform concentration dependent dielectric (ϵ) medium. It should be pointed out that due to the fact that the ionic distribution is independent of the pressure field in the limit of 1/ρ_salt,0 → 0, there is no constraint on the upper limit of ionic concentration and this can lead to unphysical results, especially at higher salt concentrations and stronger dielectric mismatch between the interior and exterior of the brush. So, one must include the effects of the local incompressibility constraint and ion-solvation to study the effects of added salt ions especially at higher salt concentrations. Results from such numerical calculations are presented in Fig. 3.

FIG. 3.

$FIG. 3. Effect of ion sizes on the distribution of (a) monomers, (b) ions, and (c) solvent. A bulk salt concentration of 0.5M was used, and all the other parameters are described in the main text. In agreement with Eq. (75), the brush exhibited shrinkage for lB0/asaltϵs > 4 (i.e., for Cl− and I−) and expansion for lB0/asaltϵs < 4 (i.e., for 2Cl− and 2I−) relative to the salt-free case [panel (a)]. The ions get partitioned between the interior of the brush and the bulk, showing more preference for the bulk due to the higher dielectric of the solvent used in these calculations. The bulk solvent volume fraction decreases with increase in the size of the salt ions. However, solvent volume fraction in the interior of the brush is qualitatively distinct from Fig. 2(c), exhibiting the role of ion-solvation in affecting solvent distribution.$

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Effect of ion sizes on the distribution of (a) monomers, (b) ions, and (c) solvent. A bulk salt concentration of 0.5M was used, and all the other parameters are described in the main text. In agreement with Eq. (75), the brush exhibited shrinkage for l_B0/a_saltϵ_s > 4 (i.e., for Cl⁻ and I⁻) and expansion for l_B0/a_saltϵ_s < 4 (i.e., for 2Cl⁻ and 2I⁻) relative to the salt-free case [panel (a)]. The ions get partitioned between the interior of the brush and the bulk, showing more preference for the bulk due to the higher dielectric of the solvent used in these calculations. The bulk solvent volume fraction decreases with increase in the size of the salt ions. However, solvent volume fraction in the interior of the brush is qualitatively distinct from Fig. 2(c), exhibiting the role of ion-solvation in affecting solvent distribution.

Figure 3 confirms the qualitative effects of ion-solvation on the monomer density profiles. In particular, the brush shrinks for Cl⁻ and I⁻ and expands for 2Cl⁻ and 2I⁻ [see panel (a) in Fig. 3], in agreement with the analytical predictions obtained from Eq. (74). Also, the salt ions get partitioned as per the dielectric of the polymer and the solvent [Fig. 3(b)]. The volume fraction of the ions in the interior is smaller than in the bulk due to the lower dielectric of the polymer used in these calculations. A rather surprising result is obtained for the solvent volume fraction distribution shown in Fig. 3(c). The bulk solvent density for different ions is exactly the same as the respective cases presented for uncharged additives in Fig. 2. However, the solvent distribution in the interior of the brush is different and highlights the importance of ion-solvation, which is significant for smaller ions, in affecting the solvent distribution. For example, the Cl⁻ exhibits a solvent volume fraction approaching the same value as for I⁻ and 2Cl⁻ at 0.5x/R_go. Figure 3 may give the perception that the effects of added salt on the monomer density profiles are insignificant. However, Eq. (75) reveals that the magnitude of the electrostatic effects can be tuned by changing either the bulk salt concentration or the valency, both of which lead to changes in κ². In order to demonstrate this, in Fig. 4, we present results showing the effects of salt concentration and valency of the ions on the monomer density profiles. In agreement with Eq. (75), the brush shrunk more with an increase in the salt concentration for the case of the Cl⁻ anion. These results are in qualitative agreement with experimental results obtained for PNIPAM and PEO brushes reported in Refs. 29, 30, 35, and 27, respectively. Similarly, an increase in the valency of the salt ions from one to two led to much stronger shrinkage due to the quadratic dependence of κ² on the valency. We have found that the brush gets desolvated along with a significant drop in salt concentration in the interior of the brush, which resembles a “salting-out” behavior of added salt on polar polymers.

FIG. 4.

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(a) Effect of salt concentration (for Cl⁻ anion) on the polymer brush, where the bulk salt concentration was varied from 0 to 2M, showing an increase in the brush shrinkage with the variation in the salt concentration. The valency of the salt, z_salt, was kept fixed at 1. (b) Effect of ion valency on the polymer brush profile showing significant collapse of the brush with an increase in valency. The salt concentration was kept fixed at 0.5M. These results are in qualitative agreement with Eq. (75).

V. CONCLUSIONS

A unified framework capturing effects of finite sizes of ions, inhomogeneous dielectric function, and ion solvation is presented in this work. Application of the theoretical framework to a planar dipolar polymer brush in equilibrium with an electrolyte solution containing symmetric ions reveals a number of interesting effects. In addition to the dielectric mismatch parameter resulting from differences in permanent dipole moments of monomers and solvents, the interplay of incompressibility, ion solvation, and translational entropy are found to be significant in affecting monomer density profiles. Crowding and ion-solvation effects can affect monomer density profiles in a significant manner. In particular, small and large ions are predicted to induce shrinkage and expansion, respectively, due to the interplay of ion-solvation and translational entropy.

The field theoretical framework developed in this work can be used in a complementary manner with other particle-based approaches such as molecular dynamics⁶⁰ and Monte-Carlo methods.⁶¹ As an example, particle-based Monte-Carlo simulations can be executed⁶¹ based on the field theoretical Hamiltonian derived in this work. Although we have considered symmetric ions here and ignored the effects of ion-binding^32–34 with the monomers, the theoretical framework presented in this work provides an exciting opportunity to study these additional important effects. For example, charge distribution^62–64 along a polymer chain due to inhomogeneous binding of the ions^41,42 can have non-trivial effects on the structure of a brush immersed in a solvent. Furthermore, ion-solvation effects have been considered by ignoring non-local dielectric effects and the dependence of the solvation energy on ion concentration. We plan to study these effects in the future in order to develop a better understanding of the effects of added salt in inhomogeneous polymeric media.

ACKNOWLEDGMENTS

This work was supported by Laboratory Directed Research and Development. The research was conducted at the Center for Nanophase Materials Sciences, which is a U.S. Department of Energy Office of Science User Facility. This research also used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

APPENDIX A: APPROXIMATE DYSON-SCHWINGER EQUATION: ${⟨δ S / δ ψ⟩}_{S_{t}} = 0$

From Eqs. (23) and (29), we can write

{⟨- \frac{1}{4 π l_{B o}} \nabla_{r}^{2} ψ (r) - \sum_{k = \pm} \frac{n_{k}}{{\bar{Q}}_{k} \{ψ, η\}} \frac{δ {\bar{Q}}_{k} \{ψ, η\}}{δ ψ (r)} - \int d r^{'} \sum_{j = p, s} ϕ_{j} (r^{'}) \frac{δ \ln γ_{j} \{ψ, r^{'}\}}{δ ψ (r)}⟩}_{S_{t}} = 0 .

(A1)

Writing $ψ (r) = \bar{ψ} (r) + Δ ψ (r)$ and using ${⟨Δ ψ (r)⟩}_{S_{t}} = 0$ ⁠,

- \frac{1}{4 π l_{B o}} \nabla_{r}^{2} \bar{ψ} (r) - \int d r^{'} \sum_{j = p, s} ϕ_{j} (r^{'}) {⟨\frac{δ \ln γ_{j} \{ψ, r^{'}\}}{δ ψ (r)}⟩}_{S_{t}} = \sum_{k = \pm} {⟨\frac{n_{k}}{{\bar{Q}}_{k} \{ψ, η\}} \frac{δ {\bar{Q}}_{k} \{ψ, η\}}{δ ψ (r)}⟩}_{S_{t}} .

(A2)

Now, we evaluate the functional derivatives and averages in this equation. We start with

{⟨\frac{n_{k}}{{\bar{Q}}_{k} \{ψ, η\}} \frac{δ {\bar{Q}}_{k} \{ψ, η\}}{δ ψ (r)}⟩}_{S_{t}} = - i z_{k} \int d r_{2} ĥ_{k} (r - r_{2}) ϕ_{k} (r_{2}),

(A3)

where we have defined

ϕ_{k} (r) = {⟨\frac{n_{k} \exp [- i \int d r^{'} ĥ_{k} (r - r^{'}) \{z_{k} ψ (r^{'}) + \frac{η (r^{'})}{ρ_{k, 0}}\}]}{\int d r^{″} \exp [- i \int d r^{'} ĥ_{k} (r^{″} - r^{'}) \{z_{k} ψ (r^{'}) + \frac{η (r^{'})}{ρ_{k, 0}}\}]}⟩}_{S_{t}}

(A4)

= \frac{n_{k} \exp [- i \int d r^{'} ĥ_{k} (r - r^{'}) \{z_{k} \bar{ψ} (r^{'}) + \frac{η (r^{'})}{ρ_{k, 0}}\} - E_{k} (r)]}{\int d r^{″} \exp [- i \int d r^{'} ĥ_{k} (r^{″} - r^{'}) \{z_{k} \bar{ψ} (r^{'}) + \frac{η (r^{'})}{ρ_{k, 0}}\} - E_{k} (r^{″})]}

(A5)

so that

\begin{matrix} E_{k} (r) = \frac{z_{k}^{2}}{2} \int d r_{1} \int d r_{2} ĥ_{k} (r - r_{1}) K (r_{1}, r_{2}, {ϕ_{j}, η}) ĥ_{k} (r - r_{2}) \end{matrix}

(A6)

and K is related to K⁻¹ by the relation

\int d r_{1} K^{- 1} (r, r_{1}, {ϕ_{j}, η}) K (r_{1}, r^{'}, {ϕ_{j}, η}) = δ (r - r^{'}) .

(A7)

See Appendix C for the Proof of Eq. (A5). The average of the functional derivative of ln γ_j in Eq. (A2) can be evaluated using Taylor’s expansion about $ψ = \bar{ψ}$ so that

\begin{matrix} \ln γ_{j} {ψ, r} & = & \ln γ_{j} {\bar{ψ}, r} + \int d r_{1} {[\frac{δ \ln γ_{j} {ψ, r}}{δ ψ (r_{1})}]}_{ψ = \bar{ψ}} Δ ψ (r_{1}) \\ + \frac{1}{2} \int d r_{1} \int d r_{2} {[\frac{δ^{2} \ln γ_{j} {ψ, r}}{δ ψ (r_{1}) δ ψ (r_{2}}]}_{ψ = \bar{ψ}} Δ ψ (r_{1}) Δ ψ (r_{2}) \end{matrix}

(A8)

which leads to

{⟨\frac{δ \ln γ_{j} {ψ, r}}{δ ψ (r^{'})}⟩}_{S_{t}} = {[\frac{δ \ln γ_{j} {ψ, r}}{δ ψ (r^{'})}]}_{ψ = \bar{ψ}}

(A9)

= \frac{p_{j}}{γ_{j} {\bar{ψ}, r}} G (| U_{j} \{ψ, r\} |) U_{j} \{ψ, r\} \cdot \nabla_{r^{'}} ĥ_{j} (r - r^{'}),

(A10)

where G(x) = (sin x/x²) (cot x − 1/x) and we have used δΔψ(r)/δψ(r′) = δ(r − r′) as well as ${⟨Δ ψ (r)⟩}_{S_{t}} = 0$ ⁠.

From Eqs. (A2), (A3), and (A10) and noting that $i \bar{ψ}$ is real,

\nabla_{r} . [\int d r_{1} ϵ (r, r_{1}) \nabla_{r_{1}} i \bar{ψ} (r_{1})] = - 4 π l_{B o} ρ_{e} (r),

(A11)

where

\begin{matrix} ϵ (r, r_{1}) & = & δ (r - r_{1}) + 4 π l_{B o} \int d r_{2} \sum_{j = p, s} p_{j}^{2} ϕ_{j} (r_{2}) ĥ_{j} (r - r_{2}) \\ \times ĥ_{j} (r_{2} - r_{1}) {\bar{G}}_{j} (| U_{j} {i \bar{ψ}, r_{2}} |), \end{matrix}

(A12)

ρ_{e} (r) = \sum_{k = \pm} z_{k} \int d r_{2} ĥ_{k} (r - r_{2}) ϕ_{k} (r_{2}),

(A13)

and we have defined

{\bar{G}}_{j} (| U_{j} {i \bar{ψ}, r_{2}} |) = \frac{L (| U_{j} {i \bar{ψ}, r_{2}} |)}{| U_{j} {i \bar{ψ}, r_{2}} |},

(A14)

where L(x) = coth x − 1/x is the Langevin function. In deriving Eq. (A11), we have used $\int d r \bar{ψ} (r) \nabla_{r} ĥ_{j} (r - r_{1}) = - \int d r ĥ_{j} (r - r_{1}) \nabla_{r} \bar{ψ} (r)$ ⁠, sin x/x = sinh(ix)/ix, and (cot x − 1/x)/x = −L(ix)/ix.

APPENDIX B: APPROXIMATE DYSON-SCHWINGER EQUATION: ${⟨ ψ δ S / δ ψ ⟩}_{S_{t}} = δ$

From Eqs. (28), (29), and (30),

{⟨\frac{δ S {ψ, ϕ_{j}, η}}{δ ψ (r)} Δ ψ (r^{'})⟩}_{S_{t}} = δ (r - r^{'}) .

(B1)

Using Eq. (23), we can write this as

- \frac{1}{4 π l_{B o}} \nabla_{r}^{2} K (r, r^{'}, {ϕ_{j}, η}) - \sum_{k = \pm} {⟨\frac{n_{k}}{{\bar{Q}}_{k} {ψ, η}} \frac{δ {\bar{Q}}_{k} \{ψ, η\}}{δ ψ (r)} Δ ψ (r^{'})⟩}_{S_{t}} - \int d r_{1} \sum_{j = p, s} ϕ_{j} (r_{1}) {⟨\frac{δ \ln γ_{j} \{ψ, r_{1}\}}{δ ψ (r)} Δ ψ (r^{'})⟩}_{S_{t}} = δ (r - r^{'}),

(B2)

where we have used ${⟨δ ψ (r) Δ ψ (r^{'})⟩}_{S_{t}} = K (r, r^{'}, {ϕ_{j}, η})$ ⁠. Other averages appearing above can be evaluated readily. In particular,

{⟨\frac{n_{k}}{{\bar{Q}}_{k} {ψ, η}} \frac{δ {\bar{Q}}_{k} \{ψ, η\}}{δ ψ (r)} Δ ψ (r^{'})⟩}_{S_{t}} = - z_{k}^{2} \int d r_{1} ĥ_{k} (r - r_{1}) ϕ_{k} (r_{1}) \int d r_{3} ĥ_{k} (r_{1} - r_{3}) K (r_{3}, r^{'}, {ϕ_{j}, η}) .

(B3)

Using Eq. (A8),

{⟨\frac{δ \ln γ_{j} \{ψ, r_{1}\}}{δ ψ (r)} Δ ψ (r^{'})⟩}_{S_{t}} = \int d r_{3} {[\frac{δ^{2} \ln γ_{j} {ψ, r_{1}}}{δ ψ (r_{3}) δ ψ (r)}]}_{ψ = \bar{ψ}} K (r_{3}, r^{'}, {ϕ_{j}, η}) .

(B4)

This leads to [cf. Eq. (38)]

\begin{matrix} \nabla_{r}^{2} K (r, r^{'}, {ϕ_{j}, η}) - 4 π l_{B o} \sum_{k = \pm} z_{k}^{2} \int d r_{1} ĥ_{k} (r - r_{1}) ϕ_{k} (r_{1}) \int d r_{3} ĥ_{k} (r_{1} - r_{3}) K (r_{3}, r^{'}, {ϕ_{j}, η}) \\ + 4 π l_{B o} \int d r_{1} \sum_{j = p, s} ϕ_{j} (r_{1}) \int d r_{3} {[\frac{δ^{2} \ln γ_{j} {ψ, r_{1}}}{δ ψ (r_{3}) δ ψ (r)}]}_{ψ = \bar{ψ}} K (r_{3}, r^{'}, {ϕ_{j}, η}) = - 4 π l_{B o} δ (r - r^{'}) . \end{matrix}

(B5)

The double derivative appearing here can be calculated using

[\frac{δ γ_{j} {ψ, r}}{δ ψ (r_{1})}] = p_{j} G (| U_{j} \{ψ, r\} |) U_{j} \{ψ, r\} \cdot \nabla_{r_{1}} ĥ_{j} (r - r_{1}),

(B6)

where G(x) = (sin x/x²) (cot x − 1/x), which leads to

[\frac{δ^{2} γ_{j} {ψ, r}}{δ ψ (r_{1}) δ ψ (r_{2})}] = p_{j}^{2} \nabla_{r_{2}} ĥ_{j} (r - r_{2}) \cdot T (U_{j} \{ψ, r\}) \cdot \nabla_{r_{1}} ĥ_{j} (r - r_{1}) .

(B7)

Here, T is a tensor, given by

\begin{matrix} T (U_{j} \{ψ, r\}) & = & [\frac{1}{| U_{j} \{ψ, r\} |} \frac{\partial G (| U_{j} \{ψ, r\} |)}{\partial | U_{j} \{ψ, r\} |}] U_{j} \{ψ, r\} U_{j} \{ψ, r\} \\ + G (| U_{j} \{ψ, r\} |) \sum_{m = x, y, z} Û_{j, m} \{ψ, r\} Û_{j, m} \{ψ, r\} \end{matrix}

(B8)

so that Û_j = U_j/|U_j|.

Furthermore, from Eq. (23),

\begin{matrix} {⟨S - S_{t}⟩}_{S_{t}} & = & - \frac{1}{8 π l_{B o}} \int d r [\bar{ψ} (r) \nabla_{r}^{2} \bar{ψ} (r)) + \nabla_{r}^{2} K (r, r, {ϕ_{j}, η})] - \sum_{k = \pm} n_{k} \ln {\hat{Q}}_{k} \{i \bar{ψ}, η\} \\ - \int d r \sum_{j = p, s} ϕ_{j} (r) [\ln γ_{j} {\bar{ψ}, r^{'}} + \frac{1}{2} \int d r_{1} \int d r_{2} {[\frac{δ^{2} \ln γ_{j} {ψ, r}}{δ ψ (r_{1}) δ ψ (r_{2})}]}_{ψ = \bar{ψ}} K (r_{1}, r_{2}, {ϕ_{j}, η})] - V δ (0) / 2, \end{matrix}

(B9)

where

{\hat{Q}}_{k} \{i \bar{ψ}\} = \frac{1}{V} \int d r^{″} \exp [- i \int d r^{'} ĥ_{k} (r^{″} - r^{'}) \{z_{k} \bar{ψ} (r^{'}) + \frac{η (r^{'})}{ρ_{k, 0}}\} - E_{k} (r^{″})] .

(B10)

Eliminating ${\hat{Q}}_{k}$ and K from Eq. (B9) using Eqs. (A5) and (B5), respectively, we get Eq. (42) from Eq. (41).

APPENDIX C: PROOF OF EQ. (A5)

With the notation, $ψ (r) = \bar{ψ} (r) + Δ ψ (r), w_{k} (r) = z_{k} \bar{ψ} (r) + \frac{η (r)}{ρ_{k, 0}}$ ⁠, we can rewrite Eq. (A4) as

ϕ_{k} (r) = \frac{n_{k} \exp [- i \int d r^{'} ĥ_{k} (r - r^{'}) w_{k} (r^{'})]}{\int d r^{″} \exp [- i \int d r^{'} ĥ_{k} (r^{″} - r^{'}) w_{k} (r^{'})]} {⟨\frac{\exp [- i z_{k} \int d r^{'} ĥ_{k} (r - r^{'}) Δ ψ (r^{'})]}{1 + g_{k} {Δ ψ}}⟩}_{S_{t}},

(C1)

where

g_{k} {Δ ψ} = \frac{\int d r_{1} \exp [- i \int d r^{'} ĥ_{k} (r_{1} - r^{'}) w_{k} (r^{'})] \{\exp [- i z_{k} \int d r^{'} ĥ_{k} (r_{1} - r^{'}) Δ ψ (r^{'})] - 1\}}{\int d r_{1} \exp [- i \int d r^{'} ĥ_{k} (r_{1} - r^{'}) w_{k} (r^{'})]} .

(C2)

We compute the averages by writing

\frac{1}{1 + g_{k} {Δ ψ}} = \sum_{m = 0}^{\infty} c_{k, m} {[g_{k} {Δ ψ}]}^{m},

(C3)

where c_k,m is coefficient of the series expansion. In particular, c_k,m = (−1)^m for |g_k{Δψ}| < 1 and the series expansion is convergent. For other cases, the series may be divergent, but it can still be summed after evaluating averages for individual terms. Explicitly,

{[g_{k} {Δ ψ}]}^{m} = \frac{\int \prod_{γ = 1}^{m} d r_{γ} \exp [- i \sum_{γ = 1}^{m} \int d r^{'} ĥ_{k} (r_{γ} - r^{'}) w_{k} (r^{'})] M_{k, m} \{Δ ψ\}}{\int \prod_{γ = 1}^{m} d r_{γ} \exp [- i \sum_{γ = 1}^{m} \int d r^{'} ĥ_{k} (r_{γ} - r^{'}) w_{k} (r^{'})]}

(C4)

so that

M_{k, m} \{Δ ψ\} = \prod_{γ = 1}^{m} \{\exp [- i z_{k} \int d r^{'} ĥ_{k} (r_{γ} - r^{'}) Δ ψ (r^{'})] - 1\},

(C5)

and we can write

\begin{matrix} {⟨\frac{\exp [- i z_{k} \int d r^{'} ĥ_{k} (r - r^{'}) Δ ψ (r^{'})]}{1 + g_{k} {Δ ψ}}⟩}_{S_{t}} & = & \sum_{m = 0}^{\infty} c_{k, m} {⟨\exp [- i z_{k} \int d r^{'} ĥ_{k} (r - r^{'}) Δ ψ (r^{'})] {[g_{k} {Δ ψ}]}^{m}⟩}_{S_{t}} \\ = & \sum_{m = 0}^{\infty} c_{k, m} \frac{\int \prod_{γ = 1}^{m} d r_{γ} \exp [- i \sum_{γ = 1}^{m} \int d r^{'} ĥ_{k} (r_{γ} - r^{'}) w_{k} (r^{'})] {⟨{\hat{M}}_{k, m} \{Δ ψ\}⟩}_{S_{t}}}{\int \prod_{γ = 1}^{m} d r_{γ} \exp [- i \sum_{γ = 1}^{m} \int d r^{'} ĥ_{k} (r_{γ} - r^{'}) w_{k} (r^{'})]} . \end{matrix}

(C6)

Here, we have defined

{⟨{\hat{M}}_{k, m} \{Δ ψ\}⟩}_{S_{t}} = {⟨\exp [- i z_{k} \int d r^{'} ĥ_{k} (r - r^{'}) Δ ψ (r^{'})] M_{k, m} \{Δ ψ\}⟩}_{S_{t}}

(C7)

= {⟨\exp [- i z_{k} \int d r^{'} ĥ_{k} (r - r^{'}) Δ ψ (r^{'})] [{(- 1)}^{m} + \sum_{γ = 1}^{m} {(- 1)}^{m - γ} (\binom{m}{γ}) \exp [- i z_{k} \sum_{γ^{'} = 1}^{γ} \int d r^{'} ĥ_{k} (r_{γ^{'}} - r^{'}) Δ ψ (r^{'})]]⟩}_{S_{t}}

(C8)

= \exp [- E_{k} (r)] [{(- 1)}^{m} + \sum_{γ = 1}^{m} {(- 1)}^{m - γ} (\binom{m}{γ}) \exp [- \sum_{γ^{'} = 1}^{γ} E_{k} (r_{γ^{'}})]] .

(C9)

In the last step, we have used ${⟨\exp [- i z_{k} \int d r^{'} ĥ_{k} (r - r^{'}) Δ ψ (r^{'})]⟩}_{S_{t}} = e^{- E_{k} (r)}$ and ${⟨\exp [- i z_{k} \sum_{γ^{'}} \int d r^{'} ĥ_{k} (r_{γ^{'}} - r^{'}) Δ ψ (r^{'})]⟩}_{S_{t}} = e^{- \sum_{γ^{'}} E_{k} (r_{γ^{'}})}$ ⁠, where E_k is given by Eq. (A6). This leads to

\begin{matrix} {⟨\frac{\exp [- i z_{k} \int d r^{'} ĥ_{k} (r - r^{'}) Δ ψ (r^{'})]}{1 + g_{k} {Δ ψ}}⟩}_{S_{t}} & = & \exp [- E_{k} (r)] \sum_{m = 0}^{\infty} c_{k, m} {[ĝ_{k} \{E_{k}\}]}^{m} \\ = & \frac{\exp [- E_{k} (r)]}{1 + ĝ_{k} \{E_{k}\}}, \end{matrix}

(C10)

where

ĝ_{k} {E_{k}} = \frac{\int d r_{1} \exp [- i \int d r^{'} ĥ_{k} (r_{1} - r^{'}) w_{k} (r^{'})] \{\exp [- E_{k} (r_{1})] - 1\}}{\int d r_{1} \exp [- i \int d r^{'} ĥ_{k} (r_{1} - r^{'}) w_{k} (r^{'})]} .

(C11)

Using Eqs. (C10) and (C11) in Eq. (C1), Eq. (A5) is obtained.

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Author(s)

Understanding the effects of symmetric salt on the structure of a planar dipolar polymer brush

I. INTRODUCTION

II. THEORY

A. Electrostatics using the Dyson-Schwinger equations

III. PLANAR POLYMER BRUSH IN EQUILIBRIUM WITH A SALTY SOLUTION

A. Symmetric salt ions: Local electroneutrality

B. Saddle-point approximation and equilibrium with the electrolyte solution

C. Laterally homogeneous brush: One-dimensional model

IV. RESULTS

A. Uncharged additives

B. Effects of ion-solvation, translational entropy, and crowding

V. CONCLUSIONS

ACKNOWLEDGMENTS

APPENDIX A: APPROXIMATE DYSON-SCHWINGER EQUATION: ${⟨δ S / δ ψ⟩}_{S_{t}} = 0$

APPENDIX B: APPROXIMATE DYSON-SCHWINGER EQUATION: ${⟨ ψ δ S / δ ψ ⟩}_{S_{t}} = δ$

APPENDIX C: PROOF OF EQ. (A5)

REFERENCES

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Understanding the effects of symmetric salt on the structure of a planar dipolar polymer brush

I. INTRODUCTION

II. THEORY

A. Electrostatics using the Dyson-Schwinger equations

III. PLANAR POLYMER BRUSH IN EQUILIBRIUM WITH A SALTY SOLUTION

A. Symmetric salt ions: Local electroneutrality

B. Saddle-point approximation and equilibrium with the electrolyte solution

C. Laterally homogeneous brush: One-dimensional model

IV. RESULTS

A. Uncharged additives

B. Effects of ion-solvation, translational entropy, and crowding

V. CONCLUSIONS

ACKNOWLEDGMENTS

APPENDIX A: APPROXIMATE DYSON-SCHWINGER EQUATION: δS/δψSt=0

APPENDIX B: APPROXIMATE DYSON-SCHWINGER EQUATION: ⟨ψδS/δψ⟩St=δ

APPENDIX C: PROOF OF EQ. (A5)

REFERENCES

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APPENDIX A: APPROXIMATE DYSON-SCHWINGER EQUATION: ${⟨δ S / δ ψ⟩}_{S_{t}} = 0$

APPENDIX B: APPROXIMATE DYSON-SCHWINGER EQUATION: ${⟨ ψ δ S / δ ψ ⟩}_{S_{t}} = δ$