Application of the Born Model to Describe Salt Partitioning in Hydrated Polymers

The classic Born model can be used to predict salt partitioning properties observed in hydrated polymers, but there are often significant quantitative discrepancies between these predictions and the experimental data. Here, we use an updated version of the Born model, reformulated to account for the local environment and mesh size of a hydrated polymer, to describe previously published NaCl, KCl, and LiCl partitioning properties of model cross-linked poly(ethylene glycol) diacrylate polymers. This reformulated Born model describes the influence of polymer structure (i.e., network mesh size and its relationship with water content) and external salt concentration on salt partitioning in the polymers with a significant improvement relative to the classic Born model. The updated model most effectively describes NaCl partitioning properties and provides an additional fundamental understanding of salt partitioning processes, for NaCl, KCl, and LiCl, in hydrated polymers that are of interest for a variety of environmental and biological applications.


Section S1. Derivation of the Electrostatic Theory
Section S1. 1

. Equilibrium condition for salt partitioning
The equilibrium condition for a hydrated polymer equilibrated with an aqueous electrolyte is: 1  ±  =  ±  Eq.S1 where  ±  and  ±  are the mean ionic activities in the polymer and external solution, respectively.
The superscript  is used to denote the polymer (i.e., membrane) phase, and the superscript  is used to denote the solution phase.Rearranging Eq.S1 and expressing the mean ionic activity in terms of the mean ionic concentration,  ± , and the mean ionic activity coefficient,  ± , yields: Eq.S2 For a 1-1 electrolyte solution, the concentration of cations and anions in the solution must be equal to maintain electronuetrality, i.e.,  ±  =    (where    defines the concentration of salt in the external solution). 3Similarly, for an uncharged polymer equilibrated with a 1-1 electrolyte,  ±  =    .Therefore, Eq.S2 can be written as: 4       =  ±   ±  Eq.S2b The salt partition (or sorption) coefficient for a hydrated polymer equilibrated with an aqueous electrolyte,   , is defined as the ratio of salt in the polymer relative to that in the external solution (i.e.,   ≡    /   ). 4 Therefore, the salt partition coefficient for an uncharged polymer equilibrated with a 1-1 electrolyte can be defined as: For an aqueous 1-1 salt, the mean ionic activity coefficient is defined using the molar excess Gibbs free energy,  ̅ ±  , as: 2,5 Eq. S4 where  and  are the gas constant and temperature, respectively, and  is an arbitrary index denoting the phase.Therefore, the ratio of activity coefficients in solution and polymer can be defined as: Eq. S5 where ∆ ̅ ±,  is the change in partial molar excess Gibbs free energy associated with the salt partitioning process, which is defined as ∆ ̅ ±,  =  ̅ ± , −  ̅ ± , .Combining Eq.S5 with Eq. S3 results in the following definition of the salt sorption coefficient of an uncharged polymer equilibrated with a 1-1 electrolyte: Eq. S6

Section S1.2. Excess solvation energy
Electrostatic Theory results from relating ∆ ̅ ±,  to experimentally measurable quantities.One approach is to relate ∆ ̅ ±,  to the mean ionic excess solvation energy as: where ∆  is the mean ionic excess solvation energy (in units of   ).In this sub-section, we outline two approaches, herein referred to as the classic Born model and Freger-Born model, to evaluate the mean ionic excess solvation energy.
The classic Born model defines the excess solvation energy of ion , ∆ ,0 , as: 2,6,7 where   is the valence of ion ,  is the elementary charge,  0 is the vacuum permittivity,   is the Boltzmann constant,   is the cavity radius for ion , and   and   are the dielectric constants of the polymer and solution, respectively.The subscript 0, introduced in Eq.S8, is provided to differentiate equations stemming from the classic Born model from those stemming from the Freger-Born model that will be differentiated subsequently with a subscript 1.The classic Born model can be used to define the mean ionic excess solvation energy (i.e., the sum of the value described by Eq.S8 for the cation and anion), 3,5 ∆ ,0 , as: Eq. S9 where  + and  − are the valences of the cation and anion, respectively, and  + and  − are the cavity radii of the cation and anion, respectively.Eq.S9 is useful in that it describes the difference in the mean ionic excess solvation energy (between polymer and solution) for any aqueous electrolyte.
For a 1-1 electrolyte (i.e.,  + =  − = 1), Eq.S9 can be simplified by defining a mean ionic average cavity radius,   : Eq. S10 The values of  + and  − are taken as the cavity radii of the ions as proposed Duignan et al. 8 The mean ionic cavity radius of NaCl, as defined by Eq.S10, is quantitatively similar to those defined by other applications of the Born model where   is defined using the geometric average of the cavity radii of the cation and anion of the salt. 2,9,10For example, the value of   calculated via Eq.S10 is 1.87 Å, and the value of   calculated using the geometric average cavity radius is 1.95 Å.
This mean ionic average cavity radius is defined so that combination of Eqs.S6, S7, S9, and S10 result in the familiar form of the Electrostatic Theory with the classic Born model for a 1-1 electrolyte: 6,7,9 Eq. S11 The difference in the mean ionic excess solvation energy between polymer and solution described by the Freger-Born model is obtained using a similar procedure as that of the classic Born model.The excess solvation energy for an ion, accounting for the interface between polymerrich and water-rich regions in the polymer matrix, ∆ ,1 , is calculated as: 6 where   is the characteristic void space of the polymer, which, for a cross-linked hydrogel polymer, can be taken as the network mesh size, and   is the dielectric constant of the water-rich void.Eq.S12 is valid for geometry where   >   . 6Using the approximation that   =   , 6 and summing Eq.S12 for the cation and anion, the mean ionic excess solvation energy becomes: Analogous to the derivation of the Electrostatic Theory with the classic Born model (Eq.S11), combining Eqs.S6, S7, and S13 results in the Electrostatic Theory with the Freger-Born model for a 1-1 electrolyte: Eq. S14 The only difference between the Electrostatic Theory with the classic Born model (Eq.S11) and the Electrostatic Theory with the Freger-Born model (Eq.S14) is that, for a 1-1 electrolyte,   is replaced by   .As such, the Electrostatic Theory with the Freger-Born model is sensitive to polymer chain configuration in a manner that is not captured by the Electrostatic Theory with the classic Born model.

Section S2. Application of the Maxwell Garnett model to hydrated polymers
The Maxwell Garnett model describes the dielectric constants of hydrated polymers experimentally measured via a dielectric relaxation spectroscopy (DRS) technique performed in the microwave frequency range. 11,12The model (Eq. 3 in the main text) has been applied to a series of cross-linked poly(glycidyl methacrylate) polymers (referred to as XL -pGMAz) by taking the dielectric constant of pure water as 80 and the dielectric constant of the dry polymer as 2.67 (estimated via a least squares regression). 13This application of the Maxwell Garnett model, when applied with the polymer taken as the continuous phase, accurately describes the relationship between water content and dielectric constant in hydrated polymers of various structures (e.g.XL -pGMAz, HEMA/GMA, and sulfonated polysulfone) (Figure S1).Alternatively, this application of the Maxwell Garnett model with water taken as the continuous phase more accurately describes the dielectric constant of commercially available hydrated Nafion 117 polymers (Figure S1).
XL -pGMAz and HEMA/GMA were cross-linked with poly(ethylene glycol) diacrylate so they are structurally similar to XLPEGDA.For this reason, we estimated the dielectric constant of the XLPEGDA using the polymer-continuous application of the Maxwell-Garnett model.For XLPEGDA, the model (Eq. 3 in the main text) was evaluated by taking the dielectric constant of water as 80 and taking the dielectric constant of the dry polymer to be 12 as previously reported. 17lculating the dielectric constant of the hydrated polymers using a polymer-continuous application of the Maxwell Garnett model requires an assumption that the phase transition from polymer to water continuity does not occur in XLPEGDA over water volumes of approximately 0.2 -0.8 (Figure 1).There is justification for this assumption in the literature because the phase transition point in heterogeneous phase models, such as the Maxwell Garnett model can be observed at remarkably dilute concentrations of the "continuous" phase.Section S3.Determining the characteristic void space size using the mesh size The effective mesh size of a cross-linked hydrogel, such as XLPEGDA, can be calculated using the polymer's structural properties when swollen in the presence of a diluent (i.e., water). 19,20e procedure for this calculation for XLPEGDA hydrogels has been previously described by Ju et al., 19 but for convenience, the procedure is summarized here.First, the average molecular weight between cross-links,   is estimated as:  Eq.S15 [23] Subsequently, the root-mean squared end-to-end distance of PEG chains between cross-links, ( 0 2 ) 1/2 , is calculated as:  For polymers with an equilibrium volume fraction of polymer greater than 0.10, Canal and Peppas suggested that the mesh size can be empirically related to the inverse equilibrium polymer volume fraction in the swollen film. 20In other words, the mesh size, , scales with the polymer water volume fraction,   , according to: ~(1 −   ) −1 . 20We used this scaling law to develop an empirical relationship between the mesh size and polymer water volume fraction in XL0, XL20, and XL40, and the XLPEGDA films synthesized by Ju et al. 19 via a least squares linear regression (Figure S2).The linear equation for the mesh size (shown on Figure S2) provides an avenue to calculate a mesh size based solely on the volume fraction of water in the polymer.This approach may be useful in situations where experimentally determined mesh sizes are not available or practical.For example, 2  =  = 5.1(1 −   ) −1 + 3.4 was used along with Eq.S14 and Eq. 3 (in the main text) to calculate the solid line in Figure 1 of the main text.

Figure S2.
Polymer mesh size reported as a function of the inverse polymer volume fraction (represented here as one minus the water volume fraction).The solid line was calculated by fitting the data using a least-squares linear regression.

Section S4. Determining the solution dielectric constant via experimental correlation models
The concentration dependent dielectric constant of aqueous electrolytes can be modeled using semi-empirical experimental correlation models, which take the general form: 26   (   ) =   + ∑       /2

𝑖=1
Eq. S18 where   is the dielectric constant of the solution,    is the concentration of the external solution,   is the dielectric constant of pure water,  is an arbitrary summation variable, and    are empirically determined parameters.The empirically determined parameters are obtained from the literature. 26The experimental correlation model for NaCl, LiCl, and KCl informed all values of solution dielectric constant for a given application of the Electrostatic Theory.polymers over a range of 0.01 -1 M concentrations. 24Similar to observations for the polymers equilibrated with NaCl, the partition coefficient of polymers equilibrated with KCl and LiCl increases with increasing solution concentration (Figure S3).The Born Model also qualitatively describes this result because the dielectric constant of the external LiCl and KCl solutions decreases as solution concentration increases (Eq. 4 in the main text), however, quantitative agreement between the updated Electrostatic Theory predictions and the experimental data is considerably worse for the polymers equilibrated with KCl and LiCl relative to those equilibrated with NaCl (Table S3).This result is consistent with the physical picture proposed in the main text where associative interactions (e.g., potassium/ethylene oxide association and lithium/water association) influence KCl and LiCl partitioning to a greater extent than NaCl partitioning.Table S3.Root mean square (RMS) log error between Electrostatic Theory predictions made with the Classic Born and Born-Freger models and experimental ion sorption data for XL0, XL20, and XL40.RMS log errors were calculated for each polymer/salt over the full range of external concentrations (0.01 M -1 M) as described previously.
For example, Morisato et al. modeled the nitrogen permeability in a blend of poly(1-trimethylsilyl-1-propyne) (PTMSP) and poly(1-phenyl-1-propyne) (PPP) using the Maxwell model and observed that the continuous phase transition point for PPP or PTMSP continuous applications of the model occurred at a volume fraction of less than 0.1 PPP.18

Table S1 .
24lymer properties used to calculate the mesh size in XL0, XL20, and XL40.24Reported by Jang et al.24  bCalculated by assuming volume additivity of polymer and water in the pre-polymerization solution. a

Table S2 .
26pirical correlation model parameters, as reported by Silva et al.,26used to calculate the concentration dependent dielectric constant of NaCl, KCl, and LiCl solutions via Eq.S18.

Concentration dependance of LiCl and KCl partitioning in XLPEGDA Jang
et al. characterized the LiCl and KCl partitioning properties of XL0, XL20, and XL40 2