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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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Sanchez-Burgos I, Muniz MC, Espinosa JR, Panagiotopoulos AZ. A Deep Potential model for liquid-vapor equilibrium and cavitation rates of water. J Chem Phys 2023; 158:2889532. [PMID: 37158636 DOI: 10.1063/5.0144500] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2023] [Accepted: 03/30/2023] [Indexed: 05/10/2023] Open
Abstract
Computational studies of liquid water and its phase transition into vapor have traditionally been performed using classical water models. Here, we utilize the Deep Potential methodology-a machine learning approach-to study this ubiquitous phase transition, starting from the phase diagram in the liquid-vapor coexistence regime. The machine learning model is trained on ab initio energies and forces based on the SCAN density functional, which has been previously shown to reproduce solid phases and other properties of water. Here, we compute the surface tension, saturation pressure, and enthalpy of vaporization for a range of temperatures spanning from 300 to 600 K and evaluate the Deep Potential model performance against experimental results and the semiempirical TIP4P/2005 classical model. Moreover, by employing the seeding technique, we evaluate the free energy barrier and nucleation rate at negative pressures for the isotherm of 296.4 K. We find that the nucleation rates obtained from the Deep Potential model deviate from those computed for the TIP4P/2005 water model due to an underestimation in the surface tension from the Deep Potential model. From analysis of the seeding simulations, we also evaluate the Tolman length for the Deep Potential water model, which is (0.091 ± 0.008) nm at 296.4 K. Finally, we identify that water molecules display a preferential orientation in the liquid-vapor interface, in which H atoms tend to point toward the vapor phase to maximize the enthalpic gain of interfacial molecules. We find that this behavior is more pronounced for planar interfaces than for the curved interfaces in bubbles. This work represents the first application of Deep Potential models to the study of liquid-vapor coexistence and water cavitation.
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Affiliation(s)
- Ignacio Sanchez-Burgos
- Maxwell Centre, Cavendish Laboratory, Department of Physics, University of Cambridge, J J Thomson Avenue,Cambridge CB3 0HE, United Kingdom
- Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Maria Carolina Muniz
- Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Jorge R Espinosa
- Maxwell Centre, Cavendish Laboratory, Department of Physics, University of Cambridge, J J Thomson Avenue,Cambridge CB3 0HE, United Kingdom
- Departamento de Química Fisica, Facultad de Ciencias Químicas, Universidad Complutense de Madrid, 28040 Madrid, Spain
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Bossert M, Trimaille I, Cagnon L, Chabaud B, Gueneau C, Spathis P, Wolf PE, Rolley E. Surface tension of cavitation bubbles. Proc Natl Acad Sci U S A 2023; 120:e2300499120. [PMID: 37023124 PMCID: PMC10104516 DOI: 10.1073/pnas.2300499120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2023] [Accepted: 03/05/2023] [Indexed: 04/07/2023] Open
Abstract
We have studied homogeneous cavitation in liquid nitrogen and normal liquid helium. We monitor the fluid content in a large number of independent mesopores with an ink-bottle shape, either when the fluid in the pores is quenched to a constant pressure or submitted to a pressure decreasing at a controlled rate. For both fluids, we show that, close enough to their critical point, the cavitation pressure threshold is in good agreement with the Classical Nucleation Theory (CNT). In contrast, at lower temperatures, deviations are observed, consistent with a reduction of the surface tension for bubbles smaller than two nanometers in radius. For nitrogen, we could accurately measure the nucleation rate as a function of the liquid pressure down to the triple point, where the critical bubble radius is about one nanometer. We find that CNT still holds, provided that the curvature dependence of the surface tension is taken into account. Furthermore, we evaluate the first- and second-order corrections in curvature, which are in reasonable agreement with recent calculations for a Lennard-Jones fluid.
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Affiliation(s)
- Marine Bossert
- Institut des NanoSciences de Paris, Sorbonne Université, CNRS, ParisF-75005, France
- Laboratoire de Physique de l’Ecole Normale Supérieure, Ecole Normale Supérieure, Université Paris Sciences et Lettres, CNRS, Sorbonne Université, Université de Paris, ParisF-75005, France
| | - I. Trimaille
- Institut des NanoSciences de Paris, Sorbonne Université, CNRS, ParisF-75005, France
| | - L. Cagnon
- Institut Néel, Université Grenoble Alpes, CNRS, GrenobleF-38042, France
| | - B. Chabaud
- Institut Néel, Université Grenoble Alpes, CNRS, GrenobleF-38042, France
| | - C. Gueneau
- Institut Néel, Université Grenoble Alpes, CNRS, GrenobleF-38042, France
| | - P. Spathis
- Institut Néel, Université Grenoble Alpes, CNRS, GrenobleF-38042, France
| | - P. E. Wolf
- Institut Néel, Université Grenoble Alpes, CNRS, GrenobleF-38042, France
| | - E. Rolley
- Laboratoire de Physique de l’Ecole Normale Supérieure, Ecole Normale Supérieure, Université Paris Sciences et Lettres, CNRS, Sorbonne Université, Université de Paris, ParisF-75005, France
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A framework for understanding the functions of biomolecular condensates across scales. Nat Rev Mol Cell Biol 2020; 22:215-235. [PMID: 33169001 DOI: 10.1038/s41580-020-00303-z] [Citation(s) in RCA: 421] [Impact Index Per Article: 105.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 10/01/2020] [Indexed: 02/07/2023]
Abstract
Biomolecular condensates are found throughout eukaryotic cells, including in the nucleus, in the cytoplasm and on membranes. They are also implicated in a wide range of cellular functions, organizing molecules that act in processes ranging from RNA metabolism to signalling to gene regulation. Early work in the field focused on identifying condensates and understanding how their physical properties and regulation arise from molecular constituents. Recent years have brought a focus on understanding condensate functions. Studies have revealed functions that span different length scales: from molecular (modulating the rates of chemical reactions) to mesoscale (organizing large structures within cells) to cellular (facilitating localization of cellular materials and homeostatic responses). In this Roadmap, we discuss representative examples of biochemical and cellular functions of biomolecular condensates from the recent literature and organize these functions into a series of non-exclusive classes across the different length scales. We conclude with a discussion of areas of current interest and challenges in the field, and thoughts about how progress may be made to further our understanding of the widespread roles of condensates in cell biology.
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Ballal D, Lu Q, Raju M, Song X. Studying vapor-liquid transition using a generalized ensemble. J Chem Phys 2019; 151:134108. [PMID: 31594333 DOI: 10.1063/1.5116252] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Homogeneous vapor-liquid nucleation is studied using the generalized Replica Exchange Method (gREM). The generalized ensemble allows the study of unstable states that cannot directly be studied in the canonical ensemble. Along with replica exchange, this allows for efficient sampling of the multiple states in a single simulation. Statistical Temperature Weighted Histogram Analysis Method is used for postprocessing to get a continuous free energy curve from bulk vapor to bulk liquid. gREM allows the study of planar, cylindrical, and spherical interfaces in a single simulation. The excess Gibbs free energy for the formation of a spherical liquid droplet in vapor for a Lennard-Jones system is calculated from the free energy curve and compared against the umbrella sampling results. The nucleation free energy barrier obtained from gREM is then used to calculate the nucleation rate without relying on any classification scheme for separating the vapor and liquid.
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Affiliation(s)
- Deepti Ballal
- Ames Laboratory, US Department of Energy, Ames, Iowa 50011, USA
| | - Qing Lu
- Ames Laboratory, US Department of Energy, Ames, Iowa 50011, USA
| | | | - Xueyu Song
- Ames Laboratory, US Department of Energy, Ames, Iowa 50011, USA and Department of Chemistry, Iowa State University, Ames, Iowa 50011, USA
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Li C, Kameyama T, Takahashi T, Kaneko T, Kato T. Nucleation dynamics of single crystal WS 2 from droplet precursors uncovered by in-situ monitoring. Sci Rep 2019; 9:12958. [PMID: 31506485 PMCID: PMC6736981 DOI: 10.1038/s41598-019-49113-0] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2019] [Accepted: 08/15/2019] [Indexed: 11/29/2022] Open
Abstract
Transition metal dichalcogenides (TMDs) attract intence attention due to its unique optoelectrical features. Recent progress in production stage of TMD enables us to synthesis uniform and large area TMD with mono layer thickness. Elucidation of growth mechanism is a challenge to improve the crystallinity of TMD, which is regargeded as a next crutial subject in the production stage. Here we report novel diffusion and nucleation dynamics during tungsten disulphide (WS2) growth. The diffusion length (Ld) of the precursors have been measured with unique nucleation control methods. It was revealed that the Ld reaches up to ~750 μm. This ultra-long diffusion can be attributed to precursor droplets observed during in-situ monitoring of WS2 growth. The integrated synthesis of >35,000 single crystals and monolayer WS2 was achieved at the wafer scale based on this model. Our findings are highly significant for both the fundamental study of droplet-mediated crystal growth and the industrial application of integrated single-crystal TMDs.
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Affiliation(s)
- Chao Li
- Department of Electronic Engineering, Tohoku University, 980-8579, Sendai, Japan
| | - Tomoya Kameyama
- Department of Electronic Engineering, Tohoku University, 980-8579, Sendai, Japan
| | - Tomoyuki Takahashi
- Department of Electronic Engineering, Tohoku University, 980-8579, Sendai, Japan
| | - Toshiro Kaneko
- Department of Electronic Engineering, Tohoku University, 980-8579, Sendai, Japan
| | - Toshiaki Kato
- Department of Electronic Engineering, Tohoku University, 980-8579, Sendai, Japan. .,JST-PRESTO, Tohoku University, 980-8579, Sendai, Japan.
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8
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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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9
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Rehner P, Gross J. Surface tension of droplets and Tolman lengths of real substances and mixtures from density functional theory. J Chem Phys 2018; 148:164703. [DOI: 10.1063/1.5020421] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Affiliation(s)
- Philipp Rehner
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany
| | - Joachim Gross
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany
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Zhukhovitskii DI. Enhancement of the droplet nucleation in a dense supersaturated Lennard-Jones vapor. J Chem Phys 2016; 144:184701. [DOI: 10.1063/1.4948436] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- D. I. Zhukhovitskii
- Joint Institute of High Temperatures, Russian Academy of Sciences, Izhorskaya 13, Bd. 2, 125412 Moscow, Russia
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12
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Bourasseau E, Malfreyt P, Ghoufi A. Surface tension and long range corrections of cylindrical interfaces. J Chem Phys 2015; 143:234708. [DOI: 10.1063/1.4937924] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023] Open
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13
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Joswiak MN, Duff N, Doherty MF, Peters B. Size-Dependent Surface Free Energy and Tolman-Corrected Droplet Nucleation of TIP4P/2005 Water. J Phys Chem Lett 2013; 4:4267-72. [PMID: 26296177 DOI: 10.1021/jz402226p] [Citation(s) in RCA: 48] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
Classical nucleation theory is notoriously inaccurate when using the macroscopic surface free energy for a planar interface. We examine the size dependence of the surface free energy for TIP4P/2005 water nanodroplets (radii ranging from 0.7 to 1.6 nm) at 300 K with the mitosis method, that is, by reversibly splitting the droplets into two subclusters. We calculate the Tolman length to be -0.56 ± 0.09 Å, which indicates that the surface free energy of water droplets that we investigated is 5-11 mJ/m(2) greater than the planar surface free energy. We incorporate the computed Tolman length into a modified classical nucleation theory (δ-CNT) and obtain modified expressions for the critical nucleus size and barrier height. δ-CNT leads to excellent agreement with independently measured nucleation kinetics.
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Affiliation(s)
- Mark N Joswiak
- †Department of Chemical Engineering and ‡Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, United States
| | - Nathan Duff
- †Department of Chemical Engineering and ‡Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, United States
| | - Michael F Doherty
- †Department of Chemical Engineering and ‡Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, United States
| | - Baron Peters
- †Department of Chemical Engineering and ‡Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, United States
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Zhukhovitskii DI. Molecular dynamics study of nanobubbles in the equilibrium Lennard-Jones fluid. J Chem Phys 2013; 139:164513. [DOI: 10.1063/1.4826648] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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15
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Processing of nano to microparticulates with controlled morphology by a novel spray pyrolysis technique: A mathematical approach to understand the process mechanism. POWDER TECHNOL 2013. [DOI: 10.1016/j.powtec.2013.02.040] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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16
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Malijevský A, Jackson G. A perspective on the interfacial properties of nanoscopic liquid drops. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2012; 24:464121. [PMID: 23114181 DOI: 10.1088/0953-8984/24/46/464121] [Citation(s) in RCA: 50] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
The structural and interfacial properties of nanoscopic liquid drops are assessed by means of mechanical, thermodynamical, and statistical mechanical approaches that are discussed in detail, including original developments at both the macroscopic level and the microscopic level of density functional theory (DFT). With a novel analysis we show that a purely macroscopic (static) mechanical treatment can lead to a qualitatively reasonable description of the surface tension and the Tolman length of a liquid drop; the latter parameter, which characterizes the curvature dependence of the tension, is found to be negative and has a magnitude of about a half of the molecular dimension. A mechanical slant cannot, however, be considered satisfactory for small finite-size systems where fluctuation effects are significant. From the opposite perspective, a curvature expansion of the macroscopic thermodynamic properties (density and chemical potential) is then used to demonstrate that a purely thermodynamic approach of this type cannot in itself correctly account for the curvature correction of the surface tension of liquid drops. We emphasize that any approach, e.g., classical nucleation theory, which is based on a purely macroscopic viewpoint, does not lead to a reliable representation when the radius of the drop becomes microscopic. The description of the enhanced inhomogeneity exhibited by small drops (particularly in the dense interior) necessitates a treatment at the molecular level to account for finite-size and surface effects correctly. The so-called mechanical route, which corresponds to a molecular-level extension of the macroscopic theory of elasticity and is particularly popular in molecular dynamics simulation, also appears to be unreliable due to the inherent ambiguity in the definition of the microscopic pressure tensor, an observation which has been known for decades but is frequently ignored. The union of the theory of capillarity (developed in the nineteenth century by Gibbs and then promoted by Tolman) with a microscopic DFT treatment allows for a direct and unambiguous description of the interfacial properties of drops of arbitrary size; DFT provides all of the bulk and surface characteristics of the system that are required to uniquely define its thermodynamic properties. In this vein, we propose a non-local mean-field DFT for Lennard-Jones (LJ) fluids to examine drops of varying size. A comparison of the predictions of our DFT with recent simulation data based on a second-order fluctuation analysis (Sampayo et al 2010 J. Chem. Phys. 132 141101) reveals the consistency of the two treatments. This observation highlights the significance of fluctuation effects in small drops, which give rise to additional entropic (thermal non-mechanical) contributions, in contrast to what one observes in the case of planar interfaces which are governed by the laws of mechanical equilibrium. A small negative Tolman length (which is found to be about a tenth of the molecular diameter) and a non-monotonic behaviour of the surface tension with the drop radius are predicted for the LJ fluid. Finally, the limits of the validity of the Tolman approach, the effect of the range of the intermolecular potential, and the behaviour of bubbles are briefly discussed.
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Affiliation(s)
- Alexandr Malijevský
- E Hála Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals of the ASCR, Prague 6, Czech Republic.
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Unn-Toc W, Halberstadt N, Meier C, Mella M. Exploring the importance of quantum effects in nucleation: the archetypical Ne(n) case. J Chem Phys 2012; 137:014304. [PMID: 22779645 DOI: 10.1063/1.4730033] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The effect of quantum mechanics (QM) on the details of the nucleation process is explored employing Ne clusters as test cases due to their semi-quantal nature. In particular, we investigate the impact of quantum mechanics on both condensation and dissociation rates in the framework of the microcanonical ensemble. Using both classical trajectories and two semi-quantal approaches (zero point averaged dynamics, ZPAD, and Gaussian-based time dependent Hartree, G-TDH) to model cluster and collision dynamics, we simulate the dissociation and monomer capture for Ne(8) as a function of the cluster internal energy, impact parameter and collision speed. The results for the capture probability P(s)(b) as a function of the impact parameter suggest that classical trajectories always underestimate capture probabilities with respect to ZPAD, albeit at most by 15%-20% in the cases we studied. They also do so in some important situations when using G-TDH. More interestingly, dissociation rates k(diss) are grossly overestimated by classical mechanics, at least by one order of magnitude. We interpret both behaviours as mainly due to the reduced amount of kinetic energy available to a quantum cluster for a chosen total internal energy. We also find that the decrease in monomer dissociation energy due to zero point energy effects plays a key role in defining dissociation rates. In fact, semi-quantal and classical results for k(diss) seem to follow a common "corresponding states" behaviour when the proper definition of internal and dissociation energies are used in a transition state model estimation of the evaporation rate constants.
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Affiliation(s)
- Wesley Unn-Toc
- Laboratoire Collisions Agrégats Réactivité-IRSAMC, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse, France
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Horsch M, Hasse H, Shchekin AK, Agarwal A, Eckelsbach S, Vrabec J, Müller EA, Jackson G. Excess equimolar radius of liquid drops. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:031605. [PMID: 22587106 DOI: 10.1103/physreve.85.031605] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2011] [Indexed: 05/31/2023]
Abstract
The curvature dependence of the surface tension is related to the excess equimolar radius of liquid drops, i.e., the deviation of the equimolar radius from the radius defined by the macroscopic capillarity approximation. Based on the Tolman [J. Chem. Phys. 17, 333 (1949)] approach and its interpretation by Nijmeijer et al. [J. Chem. Phys. 96, 565 (1991)], the surface tension of spherical interfaces is analyzed in terms of the pressure difference due to curvature. In the present study, the excess equimolar radius, which can be obtained directly from the density profile, is used instead of the Tolman length. Liquid drops of the truncated and shifted Lennard-Jones fluid are investigated by molecular dynamics simulation in the canonical ensemble, with equimolar radii ranging from 4 to 33 times the Lennard-Jones size parameter σ. In these simulations, the magnitude of the excess equimolar radius is shown to be smaller than σ/2. This suggests that the surface tension of liquid drops at the nanometer length scale is much closer to that of the planar vapor-liquid interface than reported in studies based on the mechanical route.
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Affiliation(s)
- Martin Horsch
- Lehrstuhl für Thermodynamik, Fachbereich Maschinenbau und Verfahrenstechnik, Technische Universität Kaiserslautern, Kaiserslautern, Germany.
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Tröster A, Oettel M, Block B, Virnau P, Binder K. Numerical approaches to determine the interface tension of curved interfaces from free energy calculations. J Chem Phys 2012; 136:064709. [DOI: 10.1063/1.3685221] [Citation(s) in RCA: 73] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Tröster A, Binder K. Positive Tolman length in a lattice gas with three-body interactions. PHYSICAL REVIEW LETTERS 2011; 107:265701. [PMID: 22243167 DOI: 10.1103/physrevlett.107.265701] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/14/2011] [Indexed: 05/31/2023]
Abstract
We present a new method to determine the curvature dependence of the interface tension between coexisting phases in a finite volume from free energies obtained by Monte Carlo simulations. For the example of a lattice gas on a 3D fcc lattice with nearest neighbor three-body interactions, we demonstrate how to calculate the equimolar radius R(e) as well as the radius R(s) of the surface of tension and thus the Tolman length δ(R(s))=R(e)-R(s). Within the physically relevant range of radii, δ(R(s)) shows a pronounced R(s) dependence, such that the simple Tolman parametrization for the interface tension is refutable. For the present model, extrapolation of δ(R(s)) to R(s)→∞ by various methods clearly indicates a positive limiting value.
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Affiliation(s)
- A Tröster
- Johannes Gutenberg Universität Mainz, Staudingerweg 7, D-55099 Mainz, Germany
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Corti DS, Kerr KJ, Torabi K. On the interfacial thermodynamics of nanoscale droplets and bubbles. J Chem Phys 2011; 135:024701. [PMID: 21766963 DOI: 10.1063/1.3609274] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023] Open
Abstract
We present a new self-consistent thermodynamic formalism for the interfacial properties of nanoscale embryos whose interiors do not exhibit bulklike behavior and are in complete equilibrium with the surrounding mother phase. In contrast to the standard Gibbsian analysis, whereby a bulk reference pressure based on the same temperature and chemical potentials of the mother phase is introduced, our approach naturally incorporates the normal pressure at the center of the embryo as an appropriate reference pressure. While the interfacial properties of small embryos that follow from the use of these two reference pressures are different, both methods yield by construction the same reversible work of embryo formation as well as consistency between their respective thermodynamic and mechanical routes to the surface tension. Hence, there is no a priori reason to select one method over another. Nevertheless, we argue, and demonstrate via a density-functional theory (with the local density approximation) analysis of embryo formation in the pure component Lennard-Jones fluid, that our new method generates more physically appealing trends. For example, within the new approach the surface tension at all locations of the dividing surface vanishes at the spinodal where the density profile spanning the embryo and mother phase becomes completely uniform (only the surface tension at the Gibbs surface of tension vanishes in the Gibbsian method at this same limit). Also, for bubbles, the location of the surface of tension now diverges at the spinodal, similar to the divergent behavior exhibited by the equimolar dividing surface (in the Gibbsian method, the location of the surface of tension vanishes instead). For droplets, the new method allows for the appearance of negative surface tensions (the Gibbsian method always yields positive tensions) when the normal pressures within the interior of the embryo become less than the bulk pressure of the surrounding vapor phase. Such a prediction, which is allowed by thermodynamics, is consistent with the interpretation that the mother phase's attempted compression of the droplet is counterbalanced by the negative surface tension, or free energy cost to decrease the interfacial area. Furthermore, for these same droplets, the surface of tension can no longer be meaningfully defined (the surface of tension always remains well defined in the Gibbsian method). Within the new method, the dividing surface at which the surface tension equals zero emerges as a new lengthscale, which has various thermodynamic analogs to and similar behavior as the surface of tension.
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Affiliation(s)
- David S Corti
- School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-2100, USA.
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Ghosh S, Ghosh SK. Density functional theory of size-dependent surface tension of Lennard-Jones fluid droplets using a double well type Helmholtz free energy functional. J Chem Phys 2011; 135:124710. [DOI: 10.1063/1.3633475] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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