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A special issue of Nanomaterials (ISSN 2079-4991). This special issue belongs to the section "Nanocomposite Materials".

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 29042

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Prof. Dr. Signe Kjelstrup

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PoreLab, Department of Chemistry, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Interests: non-equilibrium thermodynamic theory and its application to batteries, fuel cells, interface transport, and small systems

Special Issue Information

Dear Colleague,

This Special Issue focusses on the theory of energy conversion on the nanoscale with the aim of addressing the following questions and aspects related to them: How far down in scale can we really use classical Gibbs thermodynamics? Which theory can be used beyond the limit where classical thermodynamics ceases to apply? It is known that confinement changes the equation of the state of a fluid, but does confinement also change the equilibrium condition itself? How do we formulate the equilibrium conditions on the nanoscale?

It is now 56 years since Hill formulated his theory “Thermodynamics of Small Systems”, also called nanothermodynamics. This involved the derivation of a systematic method to deal with systems, which are no longer extensive, but it can be said that his theories have not really caught on. Therefore, development of a thermodynamic theory for the nanoscale appears to be needed.

This Special Issue aims to have a fresh look at why this is so through a review of what has happened since Hill’s method became available and by seeking to (re-)define new lines of research using this platform or choosing another platform. We invite authors to submit papers that can contribute to the advancement of nanothermodynamics using theoretical, computational, or experimental tools. The focus is on efforts that support or validate theoretical developments on the nanoscale, the scale of molecular clusters, where surface energies dominate.

Prof. Dr. Signe Kjelstrup
Guest Editor

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Keywords

thermodynamic properties of small systems
Hill’s thermodynamics
small size effects
properties of nanoporous media
clusters
curved interfaces
environmental control variables
scaling laws
confined fluids

Published Papers (11 papers)

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Editorial

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2 pages, 180 KiB

Open AccessEditorial

Special Issue on Nanoscale Thermodynamics

by Signe Kjelstrup

Nanomaterials 2021, 11(3), 584; https://doi.org/10.3390/nano11030584 - 26 Feb 2021

Cited by 1 | Viewed by 1368

Abstract

This Special Issue concerns recent developments of a theory for energy conversion on the nanoscale, namely nanothermodynamics [...] Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

Research

Jump to: Editorial

11 pages, 347 KiB

Open AccessEditor’s ChoiceArticle

Adsorption of an Ideal Gas on a Small Spherical Adsorbent

by Bjørn A. Strøm, Dick Bedeaux and Sondre K. Schnell

Nanomaterials 2021, 11(2), 431; https://doi.org/10.3390/nano11020431 - 09 Feb 2021

Cited by 5 | Viewed by 2120

Abstract

The ideal gas model is an important and useful model in classical thermodynamics. This remains so for small systems. Molecules in a gas can be adsorbed on the surface of a sphere. Both the free gas molecules and the adsorbed molecules may be [...] Read more.

U^{s}

, plays an important role in the analysis. For small adsorbents this energy depends on the curvature of the adsorbent. We model the adsorbent as a sphere with surface area

Ω = 4 π R^{2}

, where R is the radius of the sphere. We calculate the partition function for a grand canonical ensemble of two-dimensional adsorbed phases. When connected with the nanothermodynamic framework this gives us the relevant thermodynamic variables for the adsorbed phase controlled by the temperature T, surface area

Ω

, and chemical potential

μ

. The dependence of intensive variables on size may then be systematically investigated starting from the simplest model, namely the ideal adsorbed phase. This dependence is a characteristic feature of small systems which is naturally expressed by the subdivision potential of nanothermodynamics. For surface problems, the nanothermodynamic approach is different, but equivalent to Gibbs’ surface thermodynamics. It is however a general approach to the thermodynamics of small systems, and may therefore be applied to systems that do not have well defined surfaces. It is therefore desirable and useful to improve our basic understanding of nanothermodynamics. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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19 pages, 4630 KiB

Open AccessEditor’s ChoiceArticle

Nanothermodynamic Description and Molecular Simulation of a Single-Phase Fluid in a Slit Pore

by Olav Galteland, Dick Bedeaux and Signe Kjelstrup

Nanomaterials 2021, 11(1), 165; https://doi.org/10.3390/nano11010165 - 11 Jan 2021

Cited by 13 | Viewed by 2087

Abstract

We have described for the first time the thermodynamic state of a highly confined single-phase and single-component fluid in a slit pore using Hill’s thermodynamics of small systems. Hill’s theory has been named nanothermodynamics. We started by constructing an ensemble of slit pores for controlled temperature, volume, surface area, and chemical potential. We have presented the integral and differential properties according to Hill, and used them to define the disjoining pressure on the new basis. We identified all thermodynamic pressures by their mechanical counterparts in a consistent manner, and have given evidence that the identification holds true using molecular simulations. We computed the entropy and energy densities, and found in agreement with the literature, that the structures at the wall are of an energetic, not entropic nature. We have shown that the subdivision potential is unequal to zero for small wall surface areas. We have showed how Hill’s method can be used to find new Maxwell relations of a confined fluid, in addition to a scaling relation, which applies when the walls are far enough apart. By this expansion of nanothermodynamics, we have set the stage for further developments of the thermodynamics of confined fluids, a field that is central in nanotechnology. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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9 pages, 4984 KiB

Open AccessArticle

Statistical Mechanics at Strong Coupling: A Bridge between Landsberg’s Energy Levels and Hill’s Nanothermodynamics

by Rodrigo de Miguel and J. Miguel Rubí

Nanomaterials 2020, 10(12), 2471; https://doi.org/10.3390/nano10122471 - 10 Dec 2020

Cited by 8 | Viewed by 2350

Abstract

We review and show the connection between three different theories proposed for the thermodynamic treatment of systems not obeying the additivity ansatz of classical thermodynamics. In the 1950s, Landsberg proposed that when a system comes into contact with a heat bath, its energy levels are redistributed. Based on this idea, he produced an extended thermostatistical framework that accounts for unknown interactions with the environment. A decade later, Hill devised his celebrated nanothermodynamics, where he introduced the concept of subdivision potential, a new thermodynamic variable that accounts for the vanishing additivity of increasingly smaller systems. More recently, a thermostatistical framework at strong coupling has been formulated to account for the presence of the environment through a Hamiltonian of mean force. We show that this modified Hamiltonian yields a temperature-dependent energy landscape as earlier suggested by Landsberg, and it provides a thermostatistical foundation for the subdivision potential, which is the cornerstone of Hill’s nanothermodynamics. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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11 pages, 514 KiB

Open AccessEditor’s ChoiceArticle

A Legendre–Fenchel Transform for Molecular Stretching Energies

by Eivind Bering, Dick Bedeaux, Signe Kjelstrup, Astrid S. de Wijn, Ivan Latella and J. Miguel Rubi

Nanomaterials 2020, 10(12), 2355; https://doi.org/10.3390/nano10122355 - 27 Nov 2020

Cited by 6 | Viewed by 1805

Abstract

Single-molecular polymers can be used to analyze to what extent thermodynamics applies when the size of the system is drastically reduced. We have recently verified using molecular-dynamics simulations that isometric and isotensional stretching of a small polymer result in Helmholtz and Gibbs stretching energies, which are not related to a Legendre transform, as they are for sufficiently long polymers. This disparity has also been observed experimentally. Using molecular dynamics simulations of polyethylene-oxide, we document for the first time that the Helmholtz and Gibbs stretching energies can be related by a Legendre–Fenchel transform. This opens up a possibility to apply this transform to other systems which are small in Hill’s sense. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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20 pages, 641 KiB

Open AccessArticle

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

by Bjørn A. Strøm, Jianying He, Dick Bedeaux and Signe Kjelstrup

Nanomaterials 2020, 10(9), 1691; https://doi.org/10.3390/nano10091691 - 27 Aug 2020

Cited by 10 | Viewed by 2045

Abstract

Small system properties are known to depend on geometric variables in ways that are insignificant for macroscopic systems. Small system considerations are therefore usually added to the conventional description as needed. This paper presents a thermodynamic analysis of adsorbed films of any size in a systematic and general way within the framework of Hill’s nanothermodynamics. Hill showed how to deal with size and shape as variables in a systematic manner. By doing this, the common thermodynamic equations for adsorption are changed. We derived the governing thermodynamic relations characteristic of adsorption in small systems, and point out the important distinctions between these and the corresponding conventional relations for macroscopic systems. We present operational versions of the relations specialized for adsorption of gas on colloid particles, and we applied them to analyze molecular simulation data. As an illustration of their use, we report results for CO

_{2}

adsorbed on graphite spheres. We focus on the spreading pressure, and the entropy and enthalpy of adsorption, and show how the intensive properties are affected by the size of the surface, a feature specific to small systems. The subdivision potential of the film is presented for the first time, as a measure of the film’s smallness. For the system chosen, it contributes with a substantial part to the film enthalpy. This work can be considered an extension and application of the nanothermodynamic theory developed by Hill. It provides a foundation for future thermodynamic analyses of size- and shape-dependent adsorbed film systems, alternative to that presented by Gibbs. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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15 pages, 2745 KiB

Open AccessArticle

Characterizing Polymer Hydration Shell Compressibilities with the Small-System Method

by Madhusmita Tripathy, Swaminath Bharadwaj, Shadrack Jabes B. and Nico F. A. van der Vegt

Nanomaterials 2020, 10(8), 1460; https://doi.org/10.3390/nano10081460 - 25 Jul 2020

Cited by 7 | Viewed by 2645

Abstract

The small-system method (SSM) exploits the unique feature of finite-sized open systems, whose thermodynamic quantities scale with the inverse system size. This scaling enables the calculation of properties in the thermodynamic limit of macroscopic systems based on computer simulations of finite-sized systems. We herein extend the SSM to characterize the hydration shell compressibility of a generic hydrophobic polymer in water. By systematically increasing the strength of polymer-water repulsion, we find that the excess inverse thermodynamic correction factor (

Δ 1 / Γ_{s}^{\infty}

) and compressibility (

Δ χ_{s}

) of the first hydration shell change sign from negative to positive. This occurs with a concurrent decrease in water hydrogen bonding and local tetrahedral order of the hydration shell water. The crossover lengthscale corresponds to an effective polymer bead diameter of 0.7 nm and is consistent with previous works on hydration of small and large hydrophobic solutes. The crossover lengthscale in polymer hydration shell compressibility, herein identified with the SSM approach, relates to hydrophobic interactions and macromolecular conformational equilibria in aqueous solution. The SSM approach may further be applied to study thermodynamic properties of polymer solvation shells in mixed solvents. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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19 pages, 3196 KiB

Open AccessArticle

Thermodynamics of Adsorbed Methane Storage Systems Based on Peat-Derived Activated Carbons

by Ilya Men’shchikov, Andrey Shkolin, Elena Khozina and Anatoly Fomkin

Nanomaterials 2020, 10(7), 1379; https://doi.org/10.3390/nano10071379 - 15 Jul 2020

Cited by 23 | Viewed by 2932

Abstract

Two activated carbons (ACs) were prepared from peat using thermochemical K₂SO₄ activation at 1053–1133 K for 1 h, and steam activation at 1173 K for 30 (AC-4) and 45 (AC-6) min. The steam activation duration affected the microporous structure and chemical composition of ACs, which are crucial for their adsorption performance in the methane storage technique. AC-6 displays a higher micropore volume (0.60 cm³/g), specific BET surface (1334 m²/g), and a lower fraction of mesopores calculated from the benzene vapor adsorption/desorption isotherms at 293 K. Scanning electron microscopy (SEM), X-ray diffraction (XRD), and small-angle X-ray scattering (SAXS) investigations of ACs revealed their heterogeneous morphology and chemical composition determined by the precursor and activation conditions. A thermodynamic analysis of methane adsorption at pressures up to 25 MPa and temperatures from 178 to 360 K extended to impacts of the nonideality of a gaseous phase and non-inertness of an adsorbent made it possible to evaluate the heat effects and thermodynamic state functions in the methane-AC adsorption systems. At 270 K and methane adsorption value of ~8 mmol/g, the isosteric heat capacity of the methane-AC-4 system exceeded by ~45% that evaluated for the methane-AC-6 system. The higher micropore volume and structural heterogeneity of the more activated AC-6 compared to AC-4 determine its superior methane adsorption performance. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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18 pages, 6076 KiB

Open AccessArticle

Kirkwood-Buff Integrals Using Molecular Simulation: Estimation of Surface Effects

by Noura Dawass, Peter Krüger, Sondre K. Schnell, Othonas A. Moultos, Ioannis G. Economou, Thijs J. H. Vlugt and Jean-Marc Simon

Nanomaterials 2020, 10(4), 771; https://doi.org/10.3390/nano10040771 - 16 Apr 2020

Cited by 21 | Viewed by 3565

Abstract

Kirkwood-Buff (KB) integrals provide a connection between microscopic properties and thermodynamic properties of multicomponent fluids. The estimation of KB integrals using molecular simulations of finite systems requires accounting for finite size effects. In the small system method, properties of finite subvolumes with different sizes embedded in a larger volume can be used to extrapolate to macroscopic thermodynamic properties. KB integrals computed from small subvolumes scale with the inverse size of the system. This scaling was used to find KB integrals in the thermodynamic limit. To reduce numerical inaccuracies that arise from this extrapolation, alternative approaches were considered in this work. Three methods for computing KB integrals in the thermodynamic limit from information of radial distribution functions (RDFs) of finite systems were compared. These methods allowed for the computation of surface effects. KB integrals and surface terms in the thermodynamic limit were computed for Lennard–Jones (LJ) and Weeks–Chandler–Andersen (WCA) fluids. It was found that all three methods converge to the same value. The main differentiating factor was the speed of convergence with system size L. The method that required the smallest size was the one which exploited the scaling of the finite volume KB integral multiplied by L. The relationship between KB integrals and surface effects was studied for a range of densities. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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17 pages, 1269 KiB

Open AccessArticle

Two-Phase Equilibrium Conditions in Nanopores

by Michael T. Rauter, Olav Galteland, Máté Erdős, Othonas A. Moultos, Thijs J. H. Vlugt, Sondre K. Schnell, Dick Bedeaux and Signe Kjelstrup

Nanomaterials 2020, 10(4), 608; https://doi.org/10.3390/nano10040608 - 26 Mar 2020

Cited by 19 | Viewed by 3452

Abstract

It is known that thermodynamic properties of a system change upon confinement. To know how, is important for modelling of porous media. We propose to use Hill’s systematic thermodynamic analysis of confined systems to describe two-phase equilibrium in a nanopore. The integral pressure, as defined by the compression energy of a small volume, is then central. We show that the integral pressure is constant along a slit pore with a liquid and vapor in equilibrium, when Young and Young–Laplace’s laws apply. The integral pressure of a bulk fluid in a slit pore at mechanical equilibrium can be understood as the average tangential pressure inside the pore. The pressure at mechanical equilibrium, now named differential pressure, is the average of the trace of the mechanical pressure tensor divided by three as before. Using molecular dynamics simulations, we computed the integral and differential pressures,

\hat{p}

and p, respectively, analysing the data with a growing-core methodology. The value of the bulk pressure was confirmed by Gibbs ensemble Monte Carlo simulations. The pressure difference times the volume, V, is the subdivision potential of Hill,

(p - \hat{p}) V = ϵ

. The combined simulation results confirm that the integral pressure is constant along the pore, and that

ϵ / V

scales with the inverse pore width. This scaling law will be useful for prediction of thermodynamic properties of confined systems in more complicated geometries. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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12 pages, 3059 KiB

Open AccessFeature PaperArticle

Gibbs Ensemble Monte Carlo Simulation of Fluids in Confinement: Relation between the Differential and Integral Pressures

by Máté Erdős, Olav Galteland, Dick Bedeaux, Signe Kjelstrup, Othonas A. Moultos and Thijs J. H. Vlugt

Nanomaterials 2020, 10(2), 293; https://doi.org/10.3390/nano10020293 - 09 Feb 2020

Cited by 16 | Viewed by 3348

Abstract

The accurate description of the behavior of fluids in nanoporous materials is of great importance for numerous industrial applications. Recently, a new approach was reported to calculate the pressure of nanoconfined fluids. In this approach, two different pressures are defined to take into account the smallness of the system: the so-called differential and the integral pressures. Here, the effect of several factors contributing to the confinement of fluids in nanopores are investigated using the definitions of the differential and integral pressures. Monte Carlo (MC) simulations are performed in a variation of the Gibbs ensemble to study the effect of the pore geometry, fluid-wall interactions, and differential pressure of the bulk fluid phase. It is shown that the differential and integral pressure are different for small pores and become equal as the pore size increases. The ratio of the driving forces for mass transport in the bulk and in the confined fluid is also studied. It is found that, for small pore sizes (i.e., <

5 σ_{fluid}

), the ratio of the two driving forces considerably deviates from 1. Full article

(This article belongs to the Special Issue Nanoscale Thermodynamics)

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