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., $NO3\u2212$, 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 (Mg^{2+}) and calcium (Ca^{2+}) 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 Dzubiella^{34} 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.

## 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 *α*th chain. We use the notation **R**_{α}(*t*_{α}) to represent the position vector for a particular segment, *t*_{α}, along the *α*th chain. In particular, **R**_{α}(*t*_{α} = 0) = **r**_{α}(0) is the position vector for the grafted end of the *α*th chain.

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 $\u0125p$(**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 *α*th chain, an electric dipole of moment (in units of electronic charge, *e*) **p**_{α}(*t*_{α}) = *p*_{p}**u**_{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*_{s}**u**_{k} to represent the dipole moment of the *k*th 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*_{p}*N*/*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

where *n* = *n*_{p}*N* + *n*_{s} + *n*_{+} + *n*_{−} is the total number of particles and Λ is the de Broglie wavelength. *H*_{0} is the Wiener measure for the flexible polymer chains,^{47} given by

and $HwR\alpha ,r\beta $ 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. $Hw$ can be expressed using Edwards’s formulation^{47} in a form

where $wjj\u2032$ 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, $\rho ^p(r)$, $\rho ^s(r)$, $\rho ^+(r)$, and $\rho ^\u2212(r)$ represent the microscopic number density of the monomers, solvent, cations, and anions, respectively, at a certain location **r**. They are defined as

$\u0125jr\u2212r\u2032$ 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 $\u0125j$ 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 as^{41}

where *l*_{Bo} = *e*^{2}/4*πϵ*_{o}*k*_{B}*T* 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, $\rho ^e(r)=z+\rho ^+(r)+z\u2212\rho ^\u2212(r)$ is the local charge density (in units of *e*). Also, $P^ave(r\u2032)=\u222bduP^(r,u)u$, where $P^(r,u)=pp\rho \xafp(r,u)+ps\rho \xafs(r,u)$ 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, $\rho \xafp(r,u)$ and $\rho \xafs(r,u)$ are microscopic number densities of monomeric and solvent dipoles, respectively, and defined by

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

where $Zo=exp\u2212F0/kBT$ is the partition function in the absence of inter-particle interactions, given by

and we have defined

as the partition function for a single Gaussian chain anchored at **r**_{α}(0). In Eq. (10), $wp$ and $ws$ 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,

Here, we have defined *ρ*_{j}(**r**) = *∫d***r**′$\u0125j$(**r** − **r**′)*ϕ*_{j}(**r**′) along with $\chi \xafps$ (having a dimension of length^{−3}), written as

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., ($wii$*ρ*_{i,0} + $wjj$*ρ*_{j,0})/2 = $wij$*ρ*_{i,0}*ρ*_{j,0}. Moreover, $Q\xafp,\alpha wp$ is the *normalized* partition function for a polymer chain grafted at point **r**_{α}(0), given by

Similarly, $Q\xafs$ is the partition function for a single solvent molecule, given by

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

Here, we have defined^{44,45}

and

$Q\xafk=\xb1$ is the partition function for a single particle of type *k*. Explicitly, these are given by

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*,

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

Evaluating the functional derivative in Eq. (22) formally with the action $S\psi ,\varphi j,\eta $, and putting *J* = 0, leads to

where

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

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

where *S*_{t} is a trial function and chosen to be

where $\psi \xaf(r)$ and $K\u22121(r,r\u2032,\varphi j,\eta )$*are unknowns* and need to be determined by solving the approximate Dyson-Schwinger equations, written as [from Eqs. (24) and (26)]

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

where *ϵ*(**r**, **r**_{1}) is a non-local dielectric function, given by

and *ρ*_{e}(**r**) is collective local charge density, given by

Here, we have defined

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

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

where

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

where we have used $e\u2212xSt\u22431\u2212xSt\u2243e\u2212xSt$. Equation (41) can be written in a form (see Appendix B)

where $\psi \xaf$ 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 $\u0125k=\xb1\u2009=\u2009\u0125salt$, |*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 $i\psi \xaf(r)=\psi b$ as the solution of Eq. (31), where *ψ*_{b} is the electrostatic potential in the solution far from the brush. Furthermore, this leads to

so that Eq. (38) becomes

where *ϵ*_{WCL}(**r**, **r**_{3}) is a non-local dielectric function, given by

and corresponds to the weak coupling limit^{44} (WCL) of Eq. (32) (i.e., |*U*_{j}| → 0). In this work, we approximate *ϵ*_{WCL}(**r**, **r**_{3}) ≃ *ϵ*_{l}(**r**)*δ*(**r** − **r**_{3}), where

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 *τ*^{2}(**r**, **r**_{3}),

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

where

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 approximation^{46} 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

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., $\mu k=(\u2202F/\u2202nk)V,T$), we can write Eq. (55) as *F* = *F*_{ref} + Δ*F*, where *F*_{ref} = *μ*_{s}*n*_{s} + *μ*_{+}*n*_{+} + *μ*_{−}*n*_{−} ≡ *μ*_{s}*n*_{s} + *μ*_{salt}*n*_{salt}, i.e., we take the electrolyte solution without the brush as a reference. Explicitly, Δ*F* is given by

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

and $\rho sgc(r)=\u222bdr\u2032\u0125salt(r\u2212r\u2032)\varphi sgc(r\u2032)$. 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 $wp$ gives

where $q\xaf(r,N\u2212t\alpha )$ satisfies

with the condition $q\xaf(r,0)=1$ for $t\alpha \u2032=N\u2212t\alpha =0$. Similarly, $qr\alpha (r,t\alpha )$ satisfies the same equation but with the initial condition $qr\alpha (r,0)=\delta (r\u2212r\alpha )$. Furthermore, the fields, $ws$(**r**) and $wp$(**r**), can be approximated by

and

Finally, optimization with respect to the pressure field *η* gives

### 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} = *ρ*_{0} = 1/*b*^{3}. 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 $\u0125j$. The simplification results from the identity $\u0125j$(**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}.

and

where the local dielectric function of the medium (*ϵ*(*x*) ≡ *ϵ*_{l}(*x*)) is given by [cf. Eq. (47)]

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

$\rho sgc(x)\u2261\varphi sgc(x)$ and $\rho saltgc(x)\u2261\varphi saltgc(x)$ are the number densities of solvent and salt ions, respectively, at *x* [cf. Eqs. (57) and (58)], given by

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

Similarly, Eq. (59) gives

where $q\xaf(x,N\u2212t\alpha )$ satisfies

with the condition $q\xaf(x,0)=1$ for $t\alpha \u2032=N\u2212t\alpha =0$. Similarly, $qx\alpha (x,t\alpha )$ satisfies the same equation but with the initial condition $qx\alpha (x,0)=\delta (x\u2212x\alpha )$.

Equations (64)–(72) are solved self-consistently for obtaining seven different spatial variables: $i\eta (x),\u2009iws(x),iwp(x),\u2009\u03f5(x),\u2009\rho sgc(x),\u2009\rho saltgc(x)$, 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 $Rgo=(Nb2/6)1/2$. A box length of 15*R*_{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 25*R*_{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 $q\xaf$ at the substrate, i.e., $qx\alpha (x=0,t\alpha )=q\xaf(x=0,t\alpha )=0$ for all values of *t*_{α}. The grafted ends were displaced to the first grid point in the solution so that *δ* = 0.098*R*_{go} for a box length of 25*R*_{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 ($iwp$, $iws$) 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 $iwp$, $iws$ were mixed using the simple mixing scheme^{46} 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 $\Delta \u03f5=4\pi lBo(pp2\u2212ps2)\rho 0/3\u03f5s=(\u03f5p/\u03f5s\u22121)$ is a key parameter. The dipolar interactions lead to the renormalization of the polymer-solvent interaction parameter ($\chi \xafps,eff$) so that we can write

Equation (73) reveals that an effective quality of the solvent decreases with an increase in the dipolar mismatch (i.e., Δ*ϵ*) 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 $=\varphi +(x)\rho salt,0/0.6023=\varphi \u2212(x)\rho salt,0/0.6023$, where the factor of 0.6023 in the denominator is for conversion from molecules/nm^{3} to mol/l, and the numerical values for *ρ*_{salt,0} for the salts/additives are listed in Table I.

Ion . | a_{salt} (nm)
. | 1/ρ_{salt,0} (nm^{3}/molecule)
. |
---|---|---|

Cl^{−} | 0.12 | 0.0496 |

I^{−} | 0.15 | 0.0734 |

2Cl^{−} | 0.24 | 0.0992 |

2I^{−} | 0.30 | 0.1468 |

Ion . | a_{salt} (nm)
. | 1/ρ_{salt,0} (nm^{3}/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*^{1/2}2^{−1/3} to be consistent with the Gaussian definition in Ref. 57, and the molar volume of the ions was converted from cm^{3}/mol to nm^{3}/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 *ρ*_{0} = 1 nm^{−3}, *a* = 0.34 nm, $\chi \xafps/\rho 0=$ 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. $\chi \xafps/\rho 0=0.1$ 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 $a,\chi \xafps/\rho 0$ 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 $\chi \xafps$ (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 Uda^{35}, 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.

### 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

where *ϵ*(*x*) = *ϵ*_{s}(1 + Δ*ϵϕ*_{p}(*x*)/*ρ*_{0}) and $\kappa 2=8\pi lBozsalt2\rho salt,0$. $iwp$(*x*) − $iws$(*x*) is the *exchange* potential for replacing a solvent by a monomer. In the limit of Δ*ϵϕ*_{p}(*x*) → 0, an effective polymer-solvent interaction parameter can be written as

which is the prefactor for the linear term in the expansion representing $iwp$(*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 $lB0zsalt2/(asalt\u03f5s)>4$, then the polymer is expected to shrink; otherwise the polymer brush is expected to expand. $lB0zsalt2/(asalt\u03f5s)$ 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 Wang^{59} for binary polymer blends of different dielectric constants without the $\Delta \u03f5232a3$ term. In their analysis, $lB0zsalt2/asalt\u03f5$ 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.

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 0.5*x*/*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 *κ*^{2}. 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 *κ*^{2} 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.

## 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 dynamics^{60} and Monte-Carlo methods.^{61} As an example, particle-based Monte-Carlo simulations can be executed^{61} 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: $\delta S/\delta \psi St=0$

Writing $\psi (r)=\psi \xaf(r)+\Delta \psi (r)$ and using $\Delta \psi (r)St=0$,

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

where we have defined

so that

and *K* is related to *K*^{−1} by the relation

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 $\psi =\psi \xaf$ so that

which leads to

where *G*(*x*) = (sin *x*/*x*^{2}) (cot *x* − 1/*x*) and we have used *δ*Δ*ψ*(**r**)/*δψ*(**r**′) = *δ*(**r** − **r**′) as well as $\Delta \psi (r)St=0$.

where

and we have defined

where *L*(*x*) = coth *x* − 1/*x* is the Langevin function. In deriving Eq. (A11), we have used $\u222bdr\psi \xaf(r)\u2207r\u0125j(r\u2212r1)=\u2212\u222bdr\u0125j(r\u2212r1)\u2207r\psi \xaf(r)$, sin *x*/*x* = sinh(*ix*)/*ix*, and (cot *x* − 1/*x*)/*x* = −*L*(*ix*)/*ix*.

### APPENDIX B: APPROXIMATE DYSON-SCHWINGER EQUATION: $\u27e8\psi \delta S/\delta \psi \u27e9St=\delta $

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

where we have used $\delta \psi (r)\Delta \psi (r\u2032)St=K(r,r\u2032,{\varphi j,\eta})$. Other averages appearing above can be evaluated readily. In particular,

Using Eq. (A8),

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

The double derivative appearing here can be calculated using

where *G*(*x*) = (sin *x*/*x*^{2}) (cot *x* − 1/*x*), which leads to

Here, *T* is a tensor, given by

so that *Û*_{j} = *U*_{j}/|*U*_{j}|.

Furthermore, from Eq. (23),

where

### APPENDIX C: PROOF OF EQ. (A5)

With the notation, $\psi (r)=\psi \xaf(r)+\Delta \psi (r),wk(r)=zk\psi \xaf(r)+\eta (r)\rho k,0$, we can rewrite Eq. (A4) as

where

We compute the averages by writing

where *c*_{k,m} is coefficient of the series expansion. In particular, *c*_{k,m} = (−1)^{m} for |*g*_{k}{Δ*ψ*}| < 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,

so that

and we can write

Here, we have defined

In the last step, we have used $exp\u2212izk\u222bdr\u2032\u0125k(r\u2212r\u2032)\Delta \psi (r\u2032)St=e\u2212Ek(r)$ and $exp\u2212izk\u2211\gamma \u2032\u222bdr\u2032\u0125k(r\gamma \u2032\u2212r\u2032)\Delta \psi (r\u2032)St=e\u2212\u2211\gamma \u2032Ek(r\gamma \u2032)$, where *E*_{k} is given by Eq. (A6). This leads to

where