# Characterizing Polymer Hydration Shell Compressibilities with the Small-System Method

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

**r**${}_{ij}$ and

**r**${}_{ik}$. As the water molecules on the outer surface of the first hydration shell can have hydrogen bond neighbors in the second shell, those instances were included in the calculations to compute ${q}_{\mathrm{tet}}$ for the first shell as a function of $\alpha $. For comparison, ${q}_{\mathrm{tet}}$ for shells in pure water were also computed using the polymer hydration shell widths corresponding to various $\alpha $ values.

## 3. Results and Discussion

#### 3.1. Isothermal Compressibility of SPC/E Water: Sampling Fluctuations in Cylindrical Observation Volumes

#### 3.2. Thermodynamic Properties of Polymer Hydration Shells

## 4. Scope and Limitations

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TL | Thermodynamic Limit |

SSM | Small-System Method |

KBI | Kirkwood-Buff Integral |

RDF | Radial Distribution Function |

pRDF | Proximal pair correlation function |

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**Figure 1.**The polymer-water interaction potential, ${U}_{\mathrm{pw}}$, at various repulsive interaction strength, $\alpha $. The inset shows the relation between the effective size of the polymer beads, ${\sigma}_{\mathrm{p}}^{\mathrm{eff}}$, and $\alpha $.

**Figure 2.**Observation volumes used in this work to estimate thermodynamic properties. (

**a**) The isothermal compressibility of SPC/E water is estimated using cylindrical observation volumes, where the TL corresponds to both $1/L\to 0$ and $1/{r}_{\mathrm{c}}\to 0$. (

**b**) The thermodynamical quantities pertaining to polymer hydration shells are estimated using concentric cylinders as observation volume. Here, the TL corresponds to $1/L\to 0$.

**Figure 3.**(

**a**) $1/\Gamma (L,{r}_{\mathrm{c}})$ profiles for the cylindrical shells with radius ${r}_{\mathrm{c}}$ in SPC/E water at T = 300 K, as a function of $1/L$. The lines are linear fits to the data in the range L = 1 - 2 nm. (

**b**) $1/\Gamma \left({r}_{\mathrm{c}}\right)$ for the cylindrical shells as a function of $1/{r}_{\mathrm{c}}$ at T = 300 K, 360 K. The lines are linear fits to the data in the range ${r}_{\mathrm{c}}<$ 1.5 nm.

**Figure 4.**Proximal polymer-water RDFs for various strengths of repulsive interaction parameter $\alpha $.

**Figure 5.**$1/{\Gamma}_{\mathrm{s}}\left(L\right)$ profiles for the first polymer hydration shell for different values of the repulsive interaction parameter $\alpha $. The lines are linear fits to the data in the range L = 1.2–2.5 nm.

**Figure 6.**$\Delta 1/{\Gamma}_{\mathrm{s}}^{\infty}$ profiles for the first (

**a**) and second (

**b**) polymer hydration shell for various strengths $\alpha $ of the repulsive interaction. The inset in (

**a**) shows the variation of $\Delta 1/{\Gamma}_{\mathrm{s}}^{\infty}$ as a function of ${\sigma}_{\mathrm{p}}^{\mathrm{eff}}$. In (

**a**), the data points are grouped based on their variation with $\alpha $ (see text). The error bars are calculated over four distinct windows in the production trajectory. The lines are guide to the eyes.

**Figure 7.**$\Delta 1/{\Gamma}_{\mathrm{s}}\left(L\right)(=1/{\Gamma}_{\mathrm{s}}\left(L\right)-1/{\Gamma}_{\mathrm{s}}^{\u2022}\left(L\right))$ profiles for finite-sized first hydration shells with heights L = 0.4, 0.8, 2.0, and 2.5 nm as a function of the strength $\alpha $ of the repulsive interaction. The data points are grouped based on their variation with $\alpha $ (as in Figure 6). The error bars are calculated over four distinct windows in the production trajectory. The lines are guide to the eyes.

**Figure 8.**$\Delta {\chi}_{\mathrm{s}}$ profiles for the first (

**a**) and second (

**b**) polymer hydration shell for various values of the repulsive interaction parameter $\alpha $. The inset in (

**a**) shows $\Delta {\chi}_{\mathrm{s}}$ as a function of ${\sigma}_{\mathrm{p}}^{\mathrm{eff}}$. In (

**a**), the data points are grouped based on their variation with $\alpha $ (see text). The error bars are calculated over four distinct windows in the production trajectory. The lines are guide to the eyes.

**Figure 9.**(

**a**) Tetrahedral order parameter ${q}_{\mathrm{tet}}$ for the first polymer hydration shell and the corresponding shell in pure water for various strengths $\alpha $ of the repulsive interaction. (

**b**) $\Delta {q}_{\mathrm{tet}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{q}_{\mathrm{tet}}$(polymer shell) −${q}_{\mathrm{tet}}$(water shell) as a function of ${\sigma}_{\mathrm{p}}^{\mathrm{eff}}$. The error bars are calculated over four distinct windows in the production trajectory. The lines are guide to the eyes.

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**MDPI and ACS Style**

Tripathy, M.; Bharadwaj, S.; B., S.J.; van der Vegt, N.F.A. Characterizing Polymer Hydration Shell Compressibilities with the Small-System Method. *Nanomaterials* **2020**, *10*, 1460.
https://doi.org/10.3390/nano10081460

**AMA Style**

Tripathy M, Bharadwaj S, B. SJ, van der Vegt NFA. Characterizing Polymer Hydration Shell Compressibilities with the Small-System Method. *Nanomaterials*. 2020; 10(8):1460.
https://doi.org/10.3390/nano10081460

**Chicago/Turabian Style**

Tripathy, Madhusmita, Swaminath Bharadwaj, Shadrack Jabes B., and Nico F. A. van der Vegt. 2020. "Characterizing Polymer Hydration Shell Compressibilities with the Small-System Method" *Nanomaterials* 10, no. 8: 1460.
https://doi.org/10.3390/nano10081460