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Alekseechkin NV. Thermodynamic Theory of Curvature-Dependent Surface Tension: Tolman's Theory Revisited. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2024; 40:6834-6846. [PMID: 38518188 DOI: 10.1021/acs.langmuir.3c03747] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 03/24/2024]
Abstract
An exact equation for determining the Tolman length (TL) as a function of radius is obtained within the framework of classical thermodynamics and a computational procedure for solving it is proposed. As a result of implementing this procedure, the dependences of the TL and surface tension on radius are obtained for the drop and bubble cases and various equations of state. As one of the results of the thermodynamic study, a new equation for the dependence of surface tension on radius (curvature effect) alternative to the corresponding Tolman equation and associated with the spinodal point is obtained. The fundamental impossibility to determine the curvature effect analytically from the binodal point, i.e., using the Tolman equation, is established; it is calculated only from the spinodal point and is determined by the characteristics of the system at this point. The sign of the TL asymptotic value debated in the literature in recent decades is uniquely determined in the theory: it is negative for drops and positive for bubbles.
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Affiliation(s)
- Nikolay V Alekseechkin
- Akhiezer Institute for Theoretical Physics, National Science Centre "Kharkiv Institute of Physics and Technology", Akademicheskaya Street 1, Kharkiv 61108, Ukraine
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2
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Höfling F, Dietrich S. Structure of liquid-vapor interfaces: Perspectives from liquid state theory, large-scale simulations, and potential grazing-incidence x-ray diffraction. J Chem Phys 2024; 160:104107. [PMID: 38469908 DOI: 10.1063/5.0186955] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2023] [Accepted: 02/15/2024] [Indexed: 03/13/2024] Open
Abstract
Grazing-incidence x-ray diffraction (GIXRD) is a scattering technique that allows one to characterize the structure of fluid interfaces down to the molecular scale, including the measurement of surface tension and interface roughness. However, the corresponding standard data analysis at nonzero wave numbers has been criticized as to be inconclusive because the scattering intensity is polluted by the unavoidable scattering from the bulk. Here, we overcome this ambiguity by proposing a physically consistent model of the bulk contribution based on a minimal set of assumptions of experimental relevance. To this end, we derive an explicit integral expression for the background scattering, which can be determined numerically from the static structure factors of the coexisting bulk phases as independent input. Concerning the interpretation of GIXRD data inferred from computer simulations, we extend the model to account also for the finite sizes of the bulk phases, which are unavoidable in simulations. The corresponding leading-order correction beyond the dominant contribution to the scattered intensity is revealed by asymptotic analysis, which is characterized by the competition between the linear system size and the x-ray penetration depth in the case of simulations. Specifically, we have calculated the expected GIXRD intensity for scattering at the planar liquid-vapor interface of Lennard-Jones fluids with truncated pair interactions via extensive, high-precision computer simulations. The reported data cover interfacial and bulk properties of fluid states along the whole liquid-vapor coexistence line. A sensitivity analysis shows that our findings are robust with respect to the detailed definition of the mean interface position. We conclude that previous claims of an enhanced surface tension at mesoscopic scales are amenable to unambiguous tests via scattering experiments.
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Affiliation(s)
- F Höfling
- Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 6, 14195 Berlin, Germany
- Zuse Institut Berlin, Takustr. 7, 14195 Berlin, Germany
| | - S Dietrich
- Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany
- IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
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3
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Kashchiev D. Hierarchical approximations to the nucleation work in the entire range of metastability. J Chem Phys 2023; 159:094501. [PMID: 37655765 DOI: 10.1063/5.0164821] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2023] [Accepted: 08/07/2023] [Indexed: 09/02/2023] Open
Abstract
The work W to form a nucleus (also known as the critical nucleus) is a key quantity in the description of nucleation phenomena because of its exponentially strong effect on the nucleation rate. The present study provides a general approximate expression for W, which comprises a hierarchy of approximations to the dependence of W on the experimentally controlled overpressure Δp of a nucleating multicomponent phase. This general expression is used to derive explicit formulas for the lowest-order members of the W(Δp) hierarchy as well as for the respective lowest-order approximations to the Δp dependences of the nucleus surface tension, the nucleus radius, the Gibbs-Tolman length, and the stationary nucleation rate. The second-order and the third-order approximations to the W(Δp) dependence are confronted with available W(Δp) data, and the latter is found to agree very well with the data. The results obtained are applicable to homogeneous single-component or multicomponent nucleation from the binodal to the spinodal of the old phase, i.e., in the entire range of the old-phase metastability.
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Affiliation(s)
- Dimo Kashchiev
- Institute of Physical Chemistry, Bulgarian Academy of Sciences, ul. Acad. G. Bonchev 11, Sofia 1113, Bulgaria
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4
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Aasen A, Wilhelmsen Ø, Hammer M, Reguera D. Free energy of critical droplets-from the binodal to the spinodal. J Chem Phys 2023; 158:114108. [PMID: 36948791 DOI: 10.1063/5.0142533] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/03/2023] Open
Abstract
Arguably, the main challenge of nucleation theory is to accurately evaluate the work of formation of a critical embryo in the new phase, which governs the nucleation rate. In Classical Nucleation Theory (CNT), this work of formation is estimated using the capillarity approximation, which relies on the value of the planar surface tension. This approximation has been blamed for the large discrepancies between predictions from CNT and experiments. In this work, we present a study of the free energy of formation of critical clusters of the Lennard-Jones fluid truncated and shifted at 2.5σ using Monte Carlo simulations, density gradient theory, and density functional theory. We find that density gradient theory and density functional theory accurately reproduce molecular simulation results for critical droplet sizes and their free energies. The capillarity approximation grossly overestimates the free energy of small droplets. The incorporation of curvature corrections up to the second order with the Helfrich expansion greatly remedies this and performs very well for most of the experimentally accessible regions. However, it is imprecise for the smallest droplets and largest metastabilities since it does not account for a vanishing nucleation barrier at the spinodal. To remedy this, we propose a scaling function that uses all relevant ingredients without adding fitting parameters. The scaling function reproduces accurately the free energy of the formation of critical droplets for the entire metastability range and all temperatures examined and deviates from density gradient theory by less than one kBT.
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Affiliation(s)
- Ailo Aasen
- SINTEF Energy Research, NO-7465 Trondheim, Norway
| | | | | | - David Reguera
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
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5
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Bråten V, Zhang DT, Hammer M, Aasen A, Schnell SK, Wilhelmsen Ø. Equation of state for confined fluids. J Chem Phys 2022; 156:244504. [DOI: 10.1063/5.0096875] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Fluids confined in small volumes behave differently than fluids in bulk systems. For bulk systems, a compact summary of the system’s thermodynamic properties is provided by equations of state. However, there is currently a lack of successful methods to predict the thermodynamic properties of confined fluids by use of equations of state, since their thermodynamic state depends on additional parameters introduced by the enclosing surface. In this work, we present a consistent thermodynamic framework that represents an equation of state for pure, confined fluids. The total system is decomposed into a bulk phase in equilibrium with a surface phase. The equation of state is based on an existing, accurate description of the bulk fluid and uses Gibbs’ framework for surface excess properties to consistently incorporate contributions from the surface. We apply the equation of state to a Lennard-Jones spline fluid confined by a spherical surface with a Weeks–Chandler–Andersen wall-potential. The pressure and internal energy predicted from the equation of state are in good agreement with the properties obtained directly from molecular dynamics simulations. We find that when the location of the dividing surface is chosen appropriately, the properties of highly curved surfaces can be predicted from those of a planar surface. The choice of the dividing surface affects the magnitude of the surface excess properties and its curvature dependence, but the properties of the total system remain unchanged. The framework can predict the properties of confined systems with a wide range of geometries, sizes, interparticle interactions, and wall–particle interactions, and it is independent of ensemble. A targeted area of use is the prediction of thermodynamic properties in porous media, for which a possible application of the framework is elaborated.
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Affiliation(s)
- Vilde Bråten
- Department of Materials Science and Engineering, Norwegian University of Science and Technology, NTNU, Trondheim NO-7491, Norway
| | - Daniel Tianhou Zhang
- Department of Chemistry, Norwegian University of Science and Technology, NTNU, Trondheim NO-7491, Norway
| | - Morten Hammer
- PoreLab, Department of Chemistry, Norwegian University of Science and Technology, NTNU, Trondheim NO-7491, Norway
- Gas Technology, PoreLab, SINTEF Energy Research, Trondheim NO-7465, Norway
| | - Ailo Aasen
- Gas Technology, PoreLab, SINTEF Energy Research, Trondheim NO-7465, Norway
| | - Sondre Kvalvåg Schnell
- Department of Materials Science and Engineering, Norwegian University of Science and Technology, NTNU, Trondheim NO-7491, Norway
| | - Øivind Wilhelmsen
- PoreLab, Department of Chemistry, Norwegian University of Science and Technology, NTNU, Trondheim NO-7491, Norway
- Gas Technology, PoreLab, SINTEF Energy Research, Trondheim NO-7465, Norway
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6
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Wilhelmsen Ø, Aasen A. Choked liquid flow in nozzles: Crossover from heterogeneous to homogeneous cavitation and insensitivity to depressurization rate. Chem Eng Sci 2022. [DOI: 10.1016/j.ces.2021.117176] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Lulli M, Biferale L, Falcucci G, Sbragaglia M, Shan X. Mesoscale perspective on the Tolman length. Phys Rev E 2022; 105:015301. [PMID: 35193309 DOI: 10.1103/physreve.105.015301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2021] [Accepted: 12/03/2021] [Indexed: 11/07/2022]
Abstract
We demonstrate that the multiphase Shan-Chen lattice Boltzmann method (LBM) yields a curvature dependent surface tension σ as computed from three-dimensional hydrostatic droplets and bubbles simulations. Such curvature dependence is routinely characterized, at first order, by the so-called Tolman length δ. LBM allows one to precisely compute σ at the surface of tension R_{s} and determine the Tolman length from the coefficient of the first order correction. The corresponding values of δ display universality for different equations of state, following a power-law scaling near the critical temperature. The Tolman length has been studied so far mainly via computationally demanding Molecular Dynamics simulations or by means of Density Functional Theory approaches playing a pivotal role in extending Classical Nucleation Theory. The present results open a hydrodynamic-compliant mesoscale arena, in which the fundamental role of the Tolman length, alongside real-world applications to cavitation phenomena, can be effectively tackled. All the results can be independently reproduced through the "idea.deploy" framework.
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Affiliation(s)
- Matteo Lulli
- Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
| | - Luca Biferale
- Department of Physics & INFN, University of Rome "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Rome, Italy
| | - Giacomo Falcucci
- Department of Enterprise Engineering "Mario Lucertini", University of Rome "Tor Vergata", Via del Politecnico 1, 00133 Rome, Italy.,John A. Paulson School of Engineering and Applied Physics, Harvard University, 33 Oxford Street, Cambridge, Massachusetts 02138, USA
| | - Mauro Sbragaglia
- Department of Physics & INFN, University of Rome "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Rome, Italy
| | - Xiaowen Shan
- Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
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Vatin M, Duvail M, Guilbaud P, Dufrêche JF. Liquid/liquid interface in periodic boundary condition. Phys Chem Chem Phys 2021; 23:1178-1187. [PMID: 33355331 DOI: 10.1039/d0cp04629a] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We study how surface phenomena can change the interface geometry in liquid-liquid two-phase systems with periodic boundary conditions. Without any curvature effect on surface tension, planar (slab), cylindrical, and spherical structures are successively obtained as a function of the total composition and elongation of the box, in accordance with molecular dynamics simulations for a water/heptane system. The curvature effects described by Tolman relationship desymmetrize the phase diagram by stabilizing a concavity but it leads to inconsistencies with high curvature. Helfrich model partially resolves this and predicts the possible presence of shells reflecting a frustrated system.
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Affiliation(s)
- Marin Vatin
- ICSM, CEA, Univ Montpellier, CNRS, ENSCM, Marcoule, France.
| | - Magali Duvail
- ICSM, CEA, Univ Montpellier, CNRS, ENSCM, Marcoule, France.
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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:nano10091691. [PMID: 32867403 PMCID: PMC7558047 DOI: 10.3390/nano10091691] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/20/2020] [Revised: 08/18/2020] [Accepted: 08/19/2020] [Indexed: 01/07/2023]
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 CO2 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.
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Affiliation(s)
- Bjørn A. Strøm
- Department of Structural Engineering, Faculty of Engineering Science and Technology, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway;
- Correspondence:
| | - Jianying He
- Department of Structural Engineering, Faculty of Engineering Science and Technology, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway;
| | - Dick Bedeaux
- Porelab, Department of Chemistry, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway; (D.B.); (S.K.)
| | - Signe Kjelstrup
- Porelab, Department of Chemistry, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway; (D.B.); (S.K.)
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10
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Aasen A, Reguera D, Wilhelmsen Ø. Curvature Corrections Remove the Inconsistencies of Binary Classical Nucleation Theory. PHYSICAL REVIEW LETTERS 2020; 124:045701. [PMID: 32058783 DOI: 10.1103/physrevlett.124.045701] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2019] [Revised: 12/17/2019] [Indexed: 06/10/2023]
Abstract
The study of nucleation in fluid mixtures exposes challenges beyond those of pure systems. A striking example is homogeneous condensation in highly surface-active water-alcohol mixtures, where classical nucleation theory yields an unphysical, negative number of water molecules in the critical embryo. This flaw has rendered multicomponent nucleation theory useless for many industrial and scientific applications. Here, we show that this inconsistency is removed by properly incorporating the curvature dependence of the surface tension of the mixture into classical nucleation theory for multicomponent systems. The Gibbs adsorption equation is used to explain the origin of the inconsistency by linking the molecules adsorbed at the interface to the curvature corrections of the surface tension. The Tolman length and rigidity constant are determined for several water-alcohol mixtures and used to show that the corrected theory is free of physical inconsistencies and provides accurate predictions of the nucleation rates. In particular, for the ethanol-water and propanol-water mixtures, the average error in the predicted nucleation rates is reduced from 11-15 orders of magnitude to below 1.5. The curvature-corrected nucleation theory opens the door to reliable predictions of nucleation rates in multicomponent systems, which are crucial for applications ranging from atmospheric science to research on volcanos.
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Affiliation(s)
- Ailo Aasen
- Department of Energy and Process Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
- SINTEF Energy Research, NO-7465 Trondheim, Norway
| | - David Reguera
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028-Barcelona, Spain
- University of Barcelona Institute of Complex Systems (UBICS), Martí i Franquès 1, 08028 Barcelona, Spain
| | - Øivind Wilhelmsen
- Department of Energy and Process Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
- SINTEF Energy Research, NO-7465 Trondheim, Norway
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Talik A, Tarnacka M, Wojtyniak M, Kaminska E, Kaminski K, Paluch M. The influence of the nanocurvature on the surface interactions and molecular dynamics of model liquid confined in cylindrical pores. J Mol Liq 2020. [DOI: 10.1016/j.molliq.2019.111973] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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12
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Rehner P, Aasen A, Wilhelmsen Ø. Tolman lengths and rigidity constants from free-energy functionals—General expressions and comparison of theories. J Chem Phys 2019; 151:244710. [DOI: 10.1063/1.5135288] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- P. Rehner
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany
| | - A. Aasen
- Department of Energy and Process Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
- SINTEF Energy Research, NO-7465 Trondheim, Norway
| | - Ø. Wilhelmsen
- Department of Energy and Process Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
- SINTEF Energy Research, NO-7465 Trondheim, Norway
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13
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The overlapping surface layers and the disjoining pressure in a small droplet. Colloids Surf A Physicochem Eng Asp 2019. [DOI: 10.1016/j.colsurfa.2019.04.071] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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14
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Hill’s nano-thermodynamics is equivalent with Gibbs’ thermodynamics for surfaces of constant curvatures. Chem Phys Lett 2018. [DOI: 10.1016/j.cplett.2018.07.031] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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