# Phase Behavior of Ion-Containing Polymers in Polar Solvents: Predictions from a Liquid-State Theory with Local Short-Range Interactions

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## Abstract

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## 1. Introduction

## 2. Model and Methods

#### 2.1. Polymer and Solution Models

#### 2.2. Theoretical Formulation

#### 2.3. Construction of Phase Diagram

## 3. Results and Discussions

#### 3.1. GUI App for the Salt-Free Case and Selected Sample Results

#### 3.2. Effect of Chain Length and Charge Fraction

#### 3.3. Effect of Local Short-Range Interactions

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BMCSL | Boublík–Mansoori–Carnahan–Starling–Leland |

cDFT | classical Density Functional Theory |

GUI App | Graphical User Interface Application |

HPAM | partially hydrolyzed polyacrylamide |

IUPAC | International Union of Pure and Applied Chemistry |

LCST | Lower Critical Solution Temperature |

LS | Liquid State |

MSA | Mean-Spherical Approximation |

PAA | Poly(acrylic acid) |

PCEs | Polycarboxylate (ether/ester)-based Superplasticizers |

PC-SAFT | Perturbed-Chain Statistical Associating Fluid Theory |

PMAA | Poly(methacrylic acid) |

TPT1 | first-order thermodynamic perturbation theory |

TRUE | Transparent, Reproducible, Usable by others, and Extensible |

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**Figure 1.**Illustration of a partially charged polymer with total length N, a sum of the charged (${N}_{\mathrm{A}}$) segments (type A) and neutral (${N}_{\mathrm{B}}$) segments (type B) and its counterions (type C). There are short-range interactions between A and C, between A and A, and between B and B.

**Figure 2.**Examples illustrating how the charge fraction $\eta $ and sequence distribution (from top to bottom: fully charged, block copolymer, random copolymer, alternating copolymer, and block copolymer) affect the relationship between the number of bond connections among charged beads and the total number of chain segments.

**Figure 3.**Relative dielectric constant ${\u03f5}_{r}$ (left y-axis) and the corresponding Bjerrum length ${\ell}_{\mathrm{B}}$ (right y-axis) are shown as a function of temperature for liquid water at 1 atm. In preparing this figure, we used Malmberg and Maryott’s empirical model [114] of the dielectric constant of water, ${\u03f5}_{r}\left(T\right)=87.740-0.40008T+9.398\times {10}^{-4}{T}^{2}-1.410\times {10}^{-6}{T}^{3}$, where T is the temperature value in degrees Celsius (°C).

**Figure 4.**Illustration of the GUI App of a Liquid State Theory Calculator based on MATLAB (version R2022a): (

**a**) after setting values for the parameters but before clicking “Run” and (

**b**) the resulting binodal curve and critical point (shown by the filled circle). Here, the polymer solution is in a single phase below the binodal curve and is separated into two phases above the binodal curve. The App generates a Matlab “.mat” file containing the numerical results labeled by the parameters for further analysis.

**Figure 5.**Predicted binodal curves from the GUI App presented in this work. The values for the following parameters were fixed: ${N}_{\mathrm{T}}=100$, $\eta =0.5$, ${\u03f5}_{\mathrm{A}\mathrm{C}}={\u03f5}_{\mathrm{A}}=0$, ${Z}_{\mathrm{A}}=-1$, ${Z}_{\mathrm{C}}=+1$. Values for ${N}_{1}$ and ${\u03f5}_{\mathrm{B}}$ were varied, as shown in the figure. Critical points are shown as open circles.

**Figure 6.**Predicted binodal curves from the GUI App presented in this work. The values for the following parameters were fixed: ${N}_{\mathrm{T}}=100$, ${\u03f5}_{\mathrm{A}\mathrm{C}}={\u03f5}_{\mathrm{A}}={\u03f5}_{\mathrm{B}}=0$, ${Z}_{\mathrm{A}}=-1$. Values for $\eta $, ${N}_{1}$ and ${Z}_{\mathrm{C}}$ were varied, as shown in the figure. Critical points are depicted by filled symbols.

**Figure 7.**Effect of chain length: concentration of polymer ${\rho}_{\mathrm{p},\mathrm{c}}$ (left y-axis) and Bjerrum length ${\ell}_{\mathrm{B},\mathrm{c}}$ (right y-axis) of critical points in the binodal curves from the GUI App presented in this work. The values for the following parameters were fixed: ${\u03f5}_{\mathrm{A}\mathrm{C}}={\u03f5}_{\mathrm{A}}={\u03f5}_{\mathrm{B}}=0$, ${Z}_{\mathrm{A}}=-1$, ${Z}_{\mathrm{C}}=+2$, $\eta =1$. Values for ${N}_{\mathrm{T}}$ is were varied, therefore, ${N}_{1}={N}_{\mathrm{T}}\eta -1$ was varied. Symbols are numerical results.

**Figure 8.**Effect of charge fraction: (

**a**) Predicted binodal curves from the GUI App presented in this work. The values for the following parameters were fixed: ${\u03f5}_{\mathrm{A}\mathrm{C}}={\u03f5}_{\mathrm{A}}={\u03f5}_{\mathrm{B}}=0$, ${Z}_{\mathrm{A}}=-1$, ${Z}_{\mathrm{C}}=+2$, ${N}_{\mathrm{T}}={10}^{5}$, and ${N}_{1}={N}_{\mathrm{T}}\eta -1$ (block copolymer). (

**b**) Concentration of polymer ${\rho}_{\mathrm{p},\mathrm{c}}$ (left y-axis) and Bjerrum length ${\ell}_{\mathrm{B},\mathrm{c}}$ (right y-axis) of critical points in the binodal curves versus $\eta $ from 0 to 1 under different total lengths ${N}_{\mathrm{T}}$ of polymer. The values of all parameters are the same as those used in (

**a**) unless otherwise specified in the figure.

**Figure 9.**The scaling relationship: ${\ell}_{\mathrm{B},\mathrm{c}}\sim {N}_{\mathrm{T}}^{-1}$. The values for the following parameters were fixed: ${\u03f5}_{\mathrm{A}}={\u03f5}_{\mathrm{B}}=0$, ${Z}_{\mathrm{A}}=-1$ and ${Z}_{\mathrm{C}}=+2$. Values for ${N}_{\mathrm{T}}$ is were varied, therefore ${N}_{1}=1/3{N}_{\mathrm{T}}\eta -1$ was varied as well.

**Figure 10.**The scaling relationship: ${\ell}_{\mathrm{B},\mathrm{c}}\sim {\eta}^{-3/4}$. The values for the following parameters were fixed: ${N}_{T}={10}^{4}$, ${N}_{1}=k{N}_{\mathrm{T}}\eta -1$, and ${\u03f5}_{\mathrm{A}}={\u03f5}_{\mathrm{B}}={\u03f5}_{\mathrm{AC}}=0$. Values for $\eta $ is were varied, and therefore values for the ${N}_{1}$-parameter were varied as well. From top to bottom: (i) ${N}_{1}=0$ ($k\to 0$), ${\mathrm{Z}}_{\mathrm{p}}=-1$, ${\mathrm{Z}}_{\mathrm{C}}=+1$; (ii) $k=1/4$, ${\mathrm{Z}}_{\mathrm{p}}=-1$, ${\mathrm{Z}}_{\mathrm{C}}=+1$; (iii) $k=1$, ${\mathrm{Z}}_{\mathrm{p}}=-1$, ${\mathrm{Z}}_{\mathrm{C}}=+1$; (iv) $k=1$, ${\mathrm{Z}}_{\mathrm{p}}=-2$, ${\mathrm{Z}}_{\mathrm{C}}=+1$; (v) ${N}_{1}=0$, ${\mathrm{Z}}_{\mathrm{p}}=-1$, ${\mathrm{Z}}_{\mathrm{C}}=+2$.

**Figure 11.**Predicted binodal curves from the GUI App presented in this work. The values for the following parameters were fixed: ${\u03f5}_{\mathrm{A}}=0$, ${\u03f5}_{\mathrm{A}\mathrm{C}}=0$, ${N}_{\mathrm{T}}=100$, ${Z}_{\mathrm{C}}=+1$ and ${Z}_{\mathrm{A}}=-1$. Values for the charge fraction $\eta $ and the interaction strength between the neutral segments of the polymer ${\u03f5}_{\mathrm{B}}$ were varied. As $\eta $ was varied, the ${N}_{1}$-parameter, ${N}_{1}={N}_{\mathrm{T}}\eta -1$ (block copolymer), was varied as well. The dashed line shows the limiting value of ${\ell}_{\mathrm{B}}/\sigma =0$.

**Figure 12.**Bjerrum length of critical points ${\ell}_{\mathrm{B},\mathrm{c}}$ in the binodal curves. The values for the following parameters were fixed: ${\u03f5}_{\mathrm{A}}={\u03f5}_{\mathrm{B}}=0$, ${Z}_{\mathrm{A}}=-1$, ${Z}_{\mathrm{C}}=-2$, ${N}_{\mathrm{T}}={10}^{5}$. Values for $\eta $ were varied under different ${\u03f5}_{\mathrm{A}\mathrm{C}}$ and ${N}_{1}$ values.

**Figure 13.**(

**a**) Predicted binodal curves from the GUI App presented in this work. The values for the following parameters were fixed: ${\u03f5}_{\mathrm{A}\mathrm{C}}=1$, ${\u03f5}_{\mathrm{B}}=0.2$, ${\u03f5}_{\mathrm{A}}=0$, ${Z}_{\mathrm{A}}=-1$, ${Z}_{\mathrm{C}}=-2$, ${N}_{\mathrm{T}}={10}^{5}$. Values for $\eta $ were varied, therefore, ${N}_{1}=1/3{N}_{\mathrm{T}}\eta -1$ was varied as well. The blue arrow indicates the direction of increasing charged fraction ($\eta $ from 0 to $0.6$). The red hollow circles are the critical points. (

**b**) The subfigure at the bottom is a zoomed-in view showing a closer look at the binodal curves for the range of ${\ell}_{\mathrm{B}}/\sigma \in [0,\phantom{\rule{0.166667em}{0ex}}4]$ and ${\rho}_{\mathrm{p}}{\sigma}^{3}\in [0,\phantom{\rule{0.166667em}{0ex}}0.05]$.

**Figure 14.**Critical Bjerrum length (${\ell}_{\mathrm{B},\mathrm{c}}/\sigma $) for the critical points in the binodal curves. The values for the following parameters were fixed: ${\u03f5}_{\mathrm{A}}=0$, ${N}_{\mathrm{T}}=100$, ${Z}_{\mathrm{C}}=+1$ and ${Z}_{\mathrm{A}}=-1$ and (

**a**) ${\u03f5}_{\mathrm{A}\mathrm{C}}=0$, (

**b**) ${\u03f5}_{\mathrm{A}\mathrm{C}}=0.5$ and (

**c**) ${\u03f5}_{\mathrm{A}\mathrm{C}}=1.0$. Values for $\eta $ and ${\u03f5}_{\mathrm{B}}$ were varied, therefore, ${N}_{1}={N}_{\mathrm{T}}\eta -1{N}_{1}$ was varied as well. (

**d**) The 3D surfaces for (

**a**–

**c**). Notice that the maximum of ${\ell}_{\mathrm{B}}/\sigma $ shown here is ${\ell}_{\mathrm{B}}/\sigma =8$.

**Table 1.**Summary of the different species considered in this work. A = Charged segments of the polymer; B = Neutral segments of the polymer, C = Counterions of the charged segments of the polymer and salt co-ions, D = Co-ions from the added salt (assumed to be $\mathrm{C}+\mathrm{D}$).

Component | A | B | C | D |
---|---|---|---|---|

Number density | ${\rho}_{\mathrm{A}}$ | ${\rho}_{\mathrm{B}}$ | ${\rho}_{\mathrm{C}}$ | ${\rho}_{\mathrm{D}}$ |

Valence | ${Z}_{\mathrm{A}}$ | ${Z}_{\mathrm{B}}=0$ | ${Z}_{\mathrm{C}}$ | ${Z}_{\mathrm{D}}$ |

**Table 2.**Input parameters used in the GUI App. For the salt-free polymer solution, there are three types of beads (see Figure 1): Type A = Charged segments of the polymer; Type B = Neutral segments of the polymer; Type C = Counterions. The strength parameter for the dispersion interaction is introduced in Equation (5).

Notions | Definition |
---|---|

${\ell}_{B}$ max | Upper bound of the range of the Bjerrum length |

${\ell}_{B}$ min | Lower bound of the range of the Bjerrum length |

${\ell}_{B}$ step | Step length (bin size) of the Bjerrum length |

${N}_{\mathrm{T}}$ | Total number of (A+B) segments of the polymer chain |

${N}_{1}$ | Number of bond connections between charged segments (A) |

$\eta $ | Charge fraction of the polymer chain |

${\u03f5}_{\mathrm{A}\mathrm{C}}$ | Strength of dispersion interaction between A and C |

${\u03f5}_{\mathrm{A}}$ | Strength of dispersion interaction between A and A |

${\u03f5}_{\mathrm{B}}$ | Strength of dispersion interaction between B and B |

${Z}_{\mathrm{A}}$ | Valence of individual ionized groups of the polymer |

${Z}_{\mathrm{C}}$ | Valence of counterions |

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**MDPI and ACS Style**

Wang, Y.; Qiu, Q.; Yedilbayeva, A.; Kairula, D.; Dai, L.
Phase Behavior of Ion-Containing Polymers in Polar Solvents: Predictions from a Liquid-State Theory with Local Short-Range Interactions. *Polymers* **2022**, *14*, 4421.
https://doi.org/10.3390/polym14204421

**AMA Style**

Wang Y, Qiu Q, Yedilbayeva A, Kairula D, Dai L.
Phase Behavior of Ion-Containing Polymers in Polar Solvents: Predictions from a Liquid-State Theory with Local Short-Range Interactions. *Polymers*. 2022; 14(20):4421.
https://doi.org/10.3390/polym14204421

**Chicago/Turabian Style**

Wang, Yanwei, Qiyuan Qiu, Arailym Yedilbayeva, Diana Kairula, and Liang Dai.
2022. "Phase Behavior of Ion-Containing Polymers in Polar Solvents: Predictions from a Liquid-State Theory with Local Short-Range Interactions" *Polymers* 14, no. 20: 4421.
https://doi.org/10.3390/polym14204421