# When Thermodynamic Properties of Adsorbed Films Depend on Size: Fundamental Theory and Case Study

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## Abstract

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## 1. Introduction

## 2. Nanothermodynamic Framework

#### 2.1. Small vs. Large Systems

#### 2.2. Hill’s Extension

## 3. Operational Relations

#### 3.1. The Reference State

#### 3.2. Size-Dependent Thermodynamic Properties

#### 3.3. Analogue of the Gibbs Adsorption Isotherm

## 4. Methodology

#### 4.1. Simulation Techniques

#### 4.2. Data Reduction

- The isotherms were interpolated as described below and integrated using Equation (46), approximating the gas as ideal such that ${v}_{G}=kT/p$.
- Each of the resulting functions $\widehat{\phi}(p;T,\mathsf{\Omega})$, one for each area, were then evaluated at the desired pressure to give the final curve.
- The curve for the differential spreading pressure, $\phi $, was obtained from the functions $\widehat{\phi}(p;T,\mathsf{\Omega})$ for all areas, from the relation $\phi =\widehat{\phi}+\mathsf{\Omega}{(\partial \widehat{\phi}/\partial \mathsf{\Omega})}_{T,p}$, which follows from Equation (35) at constant T, p and the definition ${\epsilon}_{s}\equiv (\phi -\widehat{\phi})/\mathsf{\Omega}$. The derivative $\mathsf{\Omega}{(\partial \widehat{\phi}/\partial \mathsf{\Omega})}_{T,p}$ was approximated by $\mathsf{\Omega}{(\Delta \widehat{\phi}/\Delta \mathsf{\Omega})}_{T,p}$ which was calculated from two functions $\widehat{\phi}(p;T,{\mathsf{\Omega}}_{1})$ and $\widehat{\phi}(p;T,{\mathsf{\Omega}}_{2})$ for values of ${\mathsf{\Omega}}_{1}$ and ${\mathsf{\Omega}}_{2}$ not too far apart.
- The curve for ${\epsilon}_{s}/\mathsf{\Omega}$ was obtained as the difference $\phi -\widehat{\phi}$.
- The curves for the entropy and enthalpy were obtained by Equations (36) and (39), respectively. This required the use of Equation (46) first, so that we knew $\widehat{\phi}(p;T,\mathsf{\Omega})$. The derivative term was approximated by -$(kT/{p}_{1}){(\partial p/\partial T)}_{\widehat{\phi},\mathsf{\Omega}}\approx -(kT/{p}_{1})(\Delta p/\Delta T)$ which was calculated from two functions $p(\widehat{\phi};{T}_{1},\mathsf{\Omega})$ and $p(\widehat{\phi};{T}_{2},\mathsf{\Omega})$ for values of ${T}_{1}$ and ${T}_{2}$ not too far apart. The two temperatures used were ${T}_{1}=1.080$ and ${T}_{2}=1.165$. The functions were obtained by inversion of $\widehat{\phi}(p;{T}_{1},\mathsf{\Omega})$ and $\widehat{\phi}(p;{T}_{2},\mathsf{\Omega})$.

## 5. Results

## 6. Discussion

## 7. Conclusions and Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Film properties $\phi $, $\widehat{\phi}$ and ${\epsilon}_{s}/\mathsf{\Omega}$ for the graphite adsorbent case as functions of the adsorbent area $\mathsf{\Omega}$ at constant temperature $T=1.080$ and pressure $p=0.011$. All quantities are given in reduced units. For reference, the approximate values in SI units are $T\approx 250$ K, $p\approx 1.05$ MPa and adsorbent radii are between 6 and 25 × 10${}^{-10}$ m. The markers indicate which adsorbent sizes were simulated and the lines are there as visual aid.

**Figure 2.**Film properties enthalpy, entropy and the subdivision potential per film molecule as functions of the adsorbent area $\mathsf{\Omega}$ at constant temperature $T=1.080$ and pressure $p=0.011$. Enthalpy and entropy for a film molecule are given relative to the gas. All quantities are given in reduced units. For reference, the approximate values in SI units are $T\approx 250$ K, $p\approx 1.05$ MPa and adsorbent radii are between 6 and 25 × 10${}^{-10}$ m for the area range.

**Figure 3.**Film properties $\phi $, $\widehat{\phi}$ and ${\epsilon}_{s}/\mathsf{\Omega}$ as functions of the adsorbent area $\mathsf{\Omega}$ at constant temperature $T=1.080$ and pressure p = 1.8× 10${}^{-4}$ for the generic adsorbent case. All quantities are given in reduced units. For reference, the approximate values in SI units are $T\approx 250$ K, $p\approx 50$ kPa and adsorbent radii are between 6 and 9 × 10${}^{-10}$ m. The markers indicate which adsorbent sizes were simulated and the lines are there as a visual aid.

**Figure 4.**Film properties enthalpy, entropy and the correction function per film molecule as functions of the adsorbent area $\mathsf{\Omega}$ at constant temperature $T=1.080$ and pressure p = 1.8 × 10${}^{-4}$ for the generic adsorbent case. Enthalpy and entropy for a film molecule are given relative to the gas. All quantities are given in reduced units. For reference, the approximate values in SI units are $T\approx 250$ K, $p\approx 50$ kPa and adsorbent radii are between 6 and 9 × 10${}^{-10}$ m for the area range.

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**MDPI and ACS Style**

Strøm, B.A.; He, J.; Bedeaux, D.; Kjelstrup, S.
When Thermodynamic Properties of Adsorbed Films Depend on Size: Fundamental Theory and Case Study. *Nanomaterials* **2020**, *10*, 1691.
https://doi.org/10.3390/nano10091691

**AMA Style**

Strøm BA, He J, Bedeaux D, Kjelstrup S.
When Thermodynamic Properties of Adsorbed Films Depend on Size: Fundamental Theory and Case Study. *Nanomaterials*. 2020; 10(9):1691.
https://doi.org/10.3390/nano10091691

**Chicago/Turabian Style**

Strøm, Bjørn A., Jianying He, Dick Bedeaux, and Signe Kjelstrup.
2020. "When Thermodynamic Properties of Adsorbed Films Depend on Size: Fundamental Theory and Case Study" *Nanomaterials* 10, no. 9: 1691.
https://doi.org/10.3390/nano10091691