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Schönhöfer PWA, Marechal M, Cleaver DJ, Schröder-Turk GE. Self-assembly and entropic effects in pear-shaped colloid systems. II. Depletion attraction of pear-shaped particles in a hard-sphere solvent. J Chem Phys 2020; 153:034904. [PMID: 32716194 DOI: 10.1063/5.0007287] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023] Open
Abstract
We consider depletion effects of a pear-shaped colloidal particle in a hard-sphere solvent for two different model realizations of the pear-shaped colloidal particle. The two models are the pear hard Gaussian overlap (PHGO) particles and the hard pears of revolution (HPR). The motivation for this study is to provide a microscopic understanding for the substantially different mesoscopic self-assembly properties of these pear-shaped colloids, in dense suspensions, that have been reported in the previous studies. This is done by determining their differing depletion attractions via Monte Carlo simulations of PHGO and HPR particles in a pool of hard spheres and comparing them with excluded volume calculations of numerically obtained ideal configurations on the microscopic level. While the HPR model behaves as predicted by the analysis of excluded volumes, the PHGO model showcases a preference for splay between neighboring particles, which can be attributed to the special non-additive characteristics of the PHGO contact function. Lastly, we propose a potentially experimentally realizable pear-shaped particle model, the non-additive hard pear of revolution model, which is based on the HPR model but also features non-additive traits similar to those of PHGO particles to mimic their depletion behavior.
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Affiliation(s)
- Philipp W A Schönhöfer
- College of Science, Health, Engineering and Education, Mathematics and Statistics, Murdoch University, 90 South Street, Murdoch WA 6150, Australia
| | - Matthieu Marechal
- Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Douglas J Cleaver
- Materials and Engineering Research Institute, Sheffield Hallam University, Sheffield S1 1WB, United Kingdom
| | - Gerd E Schröder-Turk
- College of Science, Health, Engineering and Education, Mathematics and Statistics, Murdoch University, 90 South Street, Murdoch WA 6150, Australia
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Schönhöfer PWA, Marechal M, Cleaver DJ, Schröder-Turk GE. Self-assembly and entropic effects in pear-shaped colloid systems. I. Shape sensitivity of bilayer phases in colloidal pear-shaped particle systems. J Chem Phys 2020; 153:034903. [PMID: 32716179 DOI: 10.1063/5.0007286] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Abstract
The role of particle shape in self-assembly processes is a double-edged sword. On the one hand, particle shape and particle elongation are often considered the most fundamental determinants of soft matter structure formation. On the other hand, structure formation is often highly sensitive to details of shape. Here, we address the question of particle shape sensitivity for the self-assembly of hard pear-shaped particles by studying two models for this system: (a) the pear hard Gaussian overlap (PHGO) and (b) the hard pears of revolution (HPR) model. Hard pear-shaped particles, given by the PHGO model, are known to form a bicontinuous gyroid phase spontaneously. However, this model does not replicate an additive object perfectly and, hence, varies slightly in shape from a "true" pear-shape. Therefore, we investigate in the first part of this series the stability of the gyroid phase in pear-shaped particle systems. We show, based on the HPR phase diagram, that the gyroid phase does not form in pears with such a "true" hard pear-shaped potential. Moreover, we acquire first indications from the HPR and PHGO pair-correlation functions that the formation of the gyroid is probably attributed to the small non-additive properties of the PHGO potential.
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Affiliation(s)
- Philipp W A Schönhöfer
- College of Science, Health, Engineering and Education, Mathematics and Statistics, Murdoch University, 90 South Street, 6150 Murdoch, WA, Australia
| | - Matthieu Marechal
- Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Douglas J Cleaver
- Materials and Engineering Research Institute, Sheffield Hallam University, Sheffield S1 1WB, United Kingdom
| | - Gerd E Schröder-Turk
- College of Science, Health, Engineering and Education, Mathematics and Statistics, Murdoch University, 90 South Street, 6150 Murdoch, WA, Australia
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3
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Torquato S. Perspective: Basic understanding of condensed phases of matter via packing models. J Chem Phys 2018; 149:020901. [DOI: 10.1063/1.5036657] [Citation(s) in RCA: 73] [Impact Index Per Article: 12.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
Affiliation(s)
- S. Torquato
- Department of Chemistry, Department of Physics, Princeton Center for Theoretical Science, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
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Góźdź WT. Investigation of Fluid-Fluid and Solid-Solid Phase Separation of Symmetric Nonadditive Hard Spheres at High Density. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2017; 33:11727-11732. [PMID: 28826216 DOI: 10.1021/acs.langmuir.7b02228] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We have calculated the values of the critical packing fractions for the mixtures of symmetric nonadditive hard spheres at high densities for small values of the nonadditivity parameter. Calculations have been performed for solid-solid and fluid-fluid demixing transitions. A cluster algorithm for Monte Carlo simulations in a semigrand ensemble was used, and the waste recycling method was applied to improve the accuracy of the calculations. The finite size scaling analysis was employed to compute the critical packing fractions for infinite systems with high accuracy.
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Affiliation(s)
- W T Góźdź
- Institute of Physical Chemistry Polish Academy of Sciences , Kasprzaka 44/52, 01-224 Warsaw, Poland
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5
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Pellicane G, Caccamo C. A thermodynamic self-consistent theory of asymmetric hard-core Yukawa mixtures. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:414009. [PMID: 27545096 DOI: 10.1088/0953-8984/28/41/414009] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We perform structural and thermodynamic calculations in the framework of the modified hypernetted chain (MHNC) integral equation closure to the Ornstein-Zernike equation for binary mixtures of size-different particles interacting with hard-core Yukawa pair potentials. We use the Percus-Yevick (PY) bridge functions of a binary mixture of hard-sphere (HSM) particles. The hard-sphere diameters of the PY bridge functions of the HSM system are adjusted so to achieve thermodynamic consistency between the virial and compressibility equations of state. We show the benefit of thermodynamic consistency by comparing the MHNC results with the available computer simulation data reported in the literature, and we demonstrate that the self-consistent thermodynamic theory provides a better reproduction of the simulation data over other microscopic theories.
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Affiliation(s)
- Giuseppe Pellicane
- School of Chemistry and Physics, University of Kwazulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa. National Institute for Theoretical Physics (NITheP), KZN Node, Pietermaritzburg, South Africa
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Bores C, Almarza NG, Lomba E, Kahl G. Inclusions of a two dimensional fluid with competing interactions in a disordered, porous matrix. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:194127. [PMID: 25923467 DOI: 10.1088/0953-8984/27/19/194127] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The behavior of a fluid with competing interaction ranges adsorbed in a controlled pore size disordered matrix is studied by means of grand canonical Monte Carlo simulations in order to analyze the effects of confinement. The disordered matrix model is constructed from a two-dimensional non-additive hard-sphere fluid (which shows close to its demixing critical point large fluctuations in the concentration), after a subsequent quenching of the particle positions and removal of one of the components. The topology of the porous network is analyzed by means of a Delaunay tessellation procedure. The porous cavities are large enough to allow for cluster formation, which is however somewhat hindered as a result of the confinement, as seen from the comparison of cluster size distributions calculated for the fluid under confinement and in the bulk. The occurrence of lamellar phases is impeded by the disordered nature of the porous network. Analysis of two-dimensional density maps of the adsorbed fluid for given matrix configurations shows that clusters tend to build up in specific locations of the porous matrix, so as to minimize inter-cluster repulsion.
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Affiliation(s)
- Cecilia Bores
- Instituto de Química Física Rocasolano, CSIC, Calle Serrano 119, E-28026 Madrid, Spain
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Almarza NG, Martín C, Lomba E, Bores C. Demixing and confinement of non-additive hard-sphere mixtures in slit pores. J Chem Phys 2015; 142:014702. [DOI: 10.1063/1.4905273] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- N. G. Almarza
- Instituto de Química Física Rocasolano, CSIC, Serrano 119, E-28006 Madrid, Spain
| | - C. Martín
- Instituto de Química Física Rocasolano, CSIC, Serrano 119, E-28006 Madrid, Spain
| | - E. Lomba
- Instituto de Química Física Rocasolano, CSIC, Serrano 119, E-28006 Madrid, Spain
| | - C. Bores
- Instituto de Química Física Rocasolano, CSIC, Serrano 119, E-28006 Madrid, Spain
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Fantoni R, Pastore G. Monte Carlo simulation of the nonadditive restricted primitive model of ionic fluids: phase diagram and clustering. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052303. [PMID: 23767536 DOI: 10.1103/physreve.87.052303] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2012] [Revised: 03/19/2013] [Indexed: 06/02/2023]
Abstract
We report an accurate Monte Carlo calculation of the phase diagram and clustering properties of the restricted primitive model with nonadditive hard-sphere diameters. At high density the positively nonadditive fluid shows more clustering than in the additive model and the negatively nonadditive fluid shows less clustering than in the additive model; at low density the reverse scenario appears. A negative nonadditivity tends to favor the formation of neutrally charged clusters starting from the dipole. A positive nonadditivity favors the pairing of like ions at high density. The critical point of the gas-liquid phase transition moves at higher temperatures and higher densities for a negative nonadditivity and at lower temperatures and lower densities for a positive nonadditivity. The law of corresponding states does not seem to hold strictly. Our results can be used to interpret recent experimental works on room temperature ionic liquids.
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Affiliation(s)
- Riccardo Fantoni
- Dipartimento di Scienze dei Materiali e Nanosistemi, Università Ca' Foscari Venezia, Calle Larga S. Marta DD2137, I-30123 Venezia, Italy.
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Gallardo A, Grandner S, Almarza NG, Klapp SHL. Theory of repulsive charged colloids in slit-pores. J Chem Phys 2012; 137:014702. [PMID: 22779675 DOI: 10.1063/1.4730923] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/30/2023] Open
Abstract
Using classical density functional theory (DFT) we analyze the structure of the density profiles and solvation pressures of negatively charged colloids confined in slit pores. The considered model, which was already successfully employed to study a real colloidal (silica) suspension [S. H. L. Klapp et al., Phys. Rev. Lett. 100, 118303 (2008)], involves only the macroions which interact via the effective Derjaguin-Landau-Verwey-Overbeek (DLVO) potential supplemented by a hard core interaction. The solvent enters implicitly via the screening length of the DLVO interaction. The free energy functional describing the colloidal suspension consists of a hard sphere contribution obtained from fundamental measure theory and a long range contribution which is treated using two types of approximations. One of them is the mean field approximation (MFA) and the remaining is based on Rosenfeld's perturbative method for constructing the Helmholtz energy functional. These theoretical calculations are carried out at different bulk densities and wall separations to compare finally to grand canonical Monte Carlo simulations. We also consider the impact of charged walls. Our results show that the perturbative DFT method yields generally qualitatively consistent and, for some systems, also quantitatively reliable results. In MFA, on the other hand, the neglect of charge-induced correlations leads to a breakdown of this approach in a broad range of densities.
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Affiliation(s)
- Alberto Gallardo
- Instituto de Química-Física Rocasolano, CSIC, Serrano 119, E-28006 Madrid, Spain.
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Fantoni R, Santos A. Nonadditive hard-sphere fluid mixtures: a simple analytical theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:041201. [PMID: 22181125 DOI: 10.1103/physreve.84.041201] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2011] [Indexed: 05/31/2023]
Abstract
We construct a nonperturbative fully analytical approximation for the thermodynamics and the structure of nonadditive hard-sphere fluid mixtures. The method essentially lies in a heuristic extension of the Percus-Yevick solution for additive hard spheres. Extensive comparison with Monte Carlo simulation data shows a generally good agreement, especially in the case of like-like radial distribution functions.
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Affiliation(s)
- Riccardo Fantoni
- National Institute for Theoretical Physics, Stellenbosch 7600, South Africa.
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11
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ENCISO BE, ALMARZA NG, CALZAS DS, GONZALEZ MA. Low density equation of state of asymmetric hard sphere mixtures. Mol Phys 2010. [DOI: 10.1080/002689797170374] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/16/2022]
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12
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Smain F, Ould-Kaddour F. A macroion correlation effect on the structure of charged stabilized colloidal suspensions: a self-consistent integral equation study. Colloid Polym Sci 2010. [DOI: 10.1007/s00396-010-2300-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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13
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Ayadim A, Amokrane S. Binary Mixture of Nonadditive Hard Spheres Adsorbed in a Slit Pore: A Study of the Population Inversion by the Integral Equations Theory. J Phys Chem B 2010; 114:16824-31. [DOI: 10.1021/jp107157a] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023]
Affiliation(s)
- A. Ayadim
- Physique des Liquides et Milieux Complexes, Faculté des Sciences et Technologie, Université Paris-Est (Créteil), 61 Av. du Général de Gaulle, 94010 Créteil Cedex, France
| | - S. Amokrane
- Physique des Liquides et Milieux Complexes, Faculté des Sciences et Technologie, Université Paris-Est (Créteil), 61 Av. du Général de Gaulle, 94010 Créteil Cedex, France
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Santos A, López de Haro M, Yuste SB. Virial coefficients, thermodynamic properties, and fluid-fluid transition of nonadditive hard-sphere mixtures. J Chem Phys 2010; 132:204506. [PMID: 20515099 DOI: 10.1063/1.3429600] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Different theoretical approaches for the thermodynamic properties and the equation of state for multicomponent mixtures of nonadditive hard spheres in d dimensions are presented in a unified way. These include the theory by Hamad, our previous formulation, the original MIX1 theory, a recently proposed modified MIX1 theory, as well as a nonlinear extension of the MIX1 theory proposed in this paper. Explicit expressions for the compressibility factor, Helmholtz free energy, and second, third, and fourth virial coefficients are provided. A comparison is carried out with recent Monte Carlo data for the virial coefficients of asymmetric mixtures and with available simulation data for the compressibility factor, the critical consolute point, and the liquid-liquid coexistence curves. The merits and limitations of each theory are pointed out.
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Affiliation(s)
- Andrés Santos
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain.
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15
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Brunet C, Malherbe JG, Amokrane S. Binary mixture adsorbed in a slit pore: Field-induced population inversion near the bulk instability. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:021504. [PMID: 20866815 DOI: 10.1103/physreve.82.021504] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2010] [Indexed: 05/29/2023]
Abstract
The recently proposed method for modulating through an external field the composition of a binary fluid mixture adsorbed in a slit pore is discussed. The population inversion near the bulk (demixing) instability is first analyzed in the case of a symmetric mixture of nonadditive hard spheres, without field. It is next investigated for a mixture comprising dipolar particles subject to an external field. The influence of several factors on the adsorption curves including bulk composition, pore width, field direction, polarizability versus permanent dipoles, and temperature on this field induced population inversion is shown by Monte Carlo simulation.
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Affiliation(s)
- C Brunet
- Université Paris Est, Créteil, France
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16
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Lee LL, Hara MC, Simon SJ, Ramos FS, Winkle AJ, Bomont JM. Crystallization limits of the two-term Yukawa potentials based on the entropy criterion. J Chem Phys 2010; 132:074505. [DOI: 10.1063/1.3308648] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Lloyd L Lee
- Department of Chemical and Materials Engineering, California State University, Pomona, California 91768, USA.
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17
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Chung TH, Lee LL. The mean activity coefficients of 2:2 electrolyte solutions: An integral equation study of the restricted primitive model. J Chem Phys 2009; 130:134513. [DOI: 10.1063/1.3099335] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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de Miguel E, Almarza NG, Jackson G. Surface tension of the Widom-Rowlinson model. J Chem Phys 2007; 127:034707. [PMID: 17655455 DOI: 10.1063/1.2751153] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We consider the computation of the surface tension of the fluid-fluid interface for the Widom-Rowlinson [J. Chem. Phys. 52, 1670 (1970)] binary mixture from direct simulation of the inhomogeneous system. We make use of the standard mechanical route, in which the surface tension follows from the computation of the normal and tangential components of the pressure tensor of the system. In addition to the usual approach, which involves simulations of the inhomogeneous system in the canonical ensemble, we also consider the computation of the surface tension in an ensemble where the pressure perpendicular (normal) to the planar interface is kept fixed. Both approaches are seen to provide consistent values of the interfacial tension. The issue of the system-size dependence of the surface tension is addressed. In addition, simulations of the fluid-fluid coexistence properties of the mixture are performed in the semigrand canonical ensemble. Our results are compared with existing data of the Widom-Rowlinson mixture and are also examined in the light of the vapor-liquid equilibrium of the thermodynamically equivalent one-component penetrable sphere model.
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Affiliation(s)
- E de Miguel
- Departamento de Física Aplicada, Facultad de Ciencias Experimentales, Universidad de Huelva, 21071 Huelva, Spain.
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19
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Malakhov AO, Volkov VV. Phase behavior of polymer mixtures with nonadditive hard-sphere potential. POLYMER SCIENCE SERIES A 2007. [DOI: 10.1134/s0965545x0706017x] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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20
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Pellicane G, Caccamo C, Giaquinta PV, Saija F. Virial Coefficients and Demixing of Athermal Nonadditive Mixtures. J Phys Chem B 2007; 111:4503-9. [PMID: 17417901 DOI: 10.1021/jp070277m] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
We compute the fourth virial coefficient of a binary nonadditive, hard-sphere mixture over a wide range of deviations from diameter additivity and size ratios. Hinging on this knowledge, we build up a y expansion (Barboy, B.; Gelbart, W. N. J. Chem. Phys. 1979, 71, 3053) in order to trace the fluid-fluid coexistence lines, which we then compare with the available Gibbs-ensemble Monte Carlo data and with the estimates obtained through two refined integral-equation theories of the fluid state. We find that in a regime of moderately negative nonadditivity and largely asymmetric diameters, relevant to the modeling of sterically and electrostatically stabilized colloidal mixtures, the fluid-fluid critical point is unstable with respect to crystallization.
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Affiliation(s)
- G Pellicane
- Dipartimento di Fisica, Università degli Studi di Messina, Contrada Papardo, 98166 Messina, Italy.
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21
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Kofke DA. Semigrand Canonical Monte Carlo Simulation; Integration Along Coexistence Lines. ADVANCES IN CHEMICAL PHYSICS 2007. [DOI: 10.1002/9780470141649.ch13] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 03/09/2023]
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22
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Amokrane S, Ayadim A, Malherbe JG. Structure of highly asymmetric hard-sphere mixtures: An efficient closure of the Ornstein-Zernike equations. J Chem Phys 2005; 123:174508. [PMID: 16375547 DOI: 10.1063/1.2102891] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
A simple modification of the reference hypernetted chain (RHNC) closure of the multicomponent Ornstein-Zernike equations with bridge functions taken from Rosenfeld's hard-sphere bridge functional is proposed. Its main effect is to remedy the major limitation of the RHNC closure in the case of highly asymmetric mixtures--the wide domain of packing fractions in which it has no solution. The modified closure is also much faster, while being of similar complexity. This is achieved with a limited loss of accuracy, mainly for the contact value of the big sphere correlation functions. Comparison with simulation shows that inside the RHNC no-solution domain, it provides a good description of the structure, while being clearly superior to all the other closures used so far to study highly asymmetric mixtures. The generic nature of this closure and its good accuracy combined with a reduced no-solution domain open up the possibility to study the phase diagram of complex fluids beyond the hard-sphere model.
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Affiliation(s)
- S Amokrane
- Physique des Liquides et Milieux Complexes, Faculté des Sciences et de Technologie Université Paris XII, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France.
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Fenz W, Omelyan IP, Folk R. Integral equation study of an ideal Ising mixture. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:056121. [PMID: 16383702 DOI: 10.1103/physreve.72.056121] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2005] [Revised: 09/21/2005] [Indexed: 05/05/2023]
Abstract
We construct an integral equation scheme for magnetic binary mixtures of an ideal soft-core Ising fluid and a soft-sphere fluid by mapping the system onto an equivalent nonmagnetic ternary mixture. We apply the multicomponent Ornstein-Zernike equation together with a closure relation based on the soft mean spherical approximation and a field constraint for the Ising fluid component. Phase coexistence curves are calculated both by directly evaluating the chemical potentials via the bridge function, and by using a Maxwell-like construction which is derived in the text. Our results are compared to Monte Carlo data obtained earlier, and we find that the second method yields a much better agreement with the simulations.
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Affiliation(s)
- W Fenz
- Institute for Theoretical Physics, Linz University, A-4040 Linz, Austria
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Wilson DS, Lee LL. Chemical potentials and phase equilibria of Lennard-Jones mixtures: a self-consistent integral equation approach. J Chem Phys 2005; 123:044512. [PMID: 16095374 DOI: 10.1063/1.1961399] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We explore the vapor-liquid phase behavior of binary mixtures of Lennard-Jones-type molecules where one component is supercritical, given the system temperature. We apply the self-consistency approach to the Ornstein-Zernike integral equations to obtain the correlation functions. The consistency checks include not only thermodynamic consistencies (pressure consistency and Gibbs-Duhem consistency), but also pointwise consistencies, such as the zero-separation theorems on the cavity functions. The consistencies are enforced via the bridge functions in the closure which contain adjustable parameters. The full solution requires the values of not only the monomer chemical potentials, but also the dimer chemical potentials present in the zero-separation theorems. These are evaluated by the direct chemical-potential formula [L. L. Lee, J. Chem. Phys. 97, 8606 (1992)] that does not require temperature nor density integration. In order to assess the integral equation accuracy, molecular-dynamics simulations are carried out alongside the states studied. The integral equation results compare well with simulation data. In phase calculations, it is important to have pressure consistency and valid chemical potentials, since the matching of phase boundaries requires the equality of the pressures and chemical potentials of both the liquid and vapor phases. The mixtures studied are methane-type and pentane-type molecules, both characterized by effective Lennard-Jones potentials. Calculations on one isotherm show that the integral equation approach yields valid answers as compared with the experimental data of Sage and Lacey. To study vapor-liquid phase behavior, it is necessary to use consistent theories; any inconsistencies, especially in pressure, will vitiate the phase boundary calculations.
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Affiliation(s)
- D Scott Wilson
- School of Chemical, Biological, and Materials Engineering, University of Oklahoma, Norman, Oklahoma 73072, USA
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Pastore G, Santin R, Taraphder S, Colonna F. Fluid-phase diagrams of binary mixtures from constant pressure integral equations. J Chem Phys 2005; 122:181104. [PMID: 15918685 DOI: 10.1063/1.1915347] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A new algorithm for solving integral equations of the theory of liquids at fixed pressure is introduced. Combining this technique with the Lee's star function approximation for the chemical potentials, we obtain an efficient method to investigate fluid-phase diagrams of binary mixtures. We have tested the capabilities of such technique to study symmetric and asymmetric phase diagrams in nonadditive hard spheres and Lennard-Jones mixtures. We find that the integral equation theories, although approximate, can provide a flexible tool to determine the fluid-phase diagrams whose accuracy is critically dependent on the quality of the closure and of the resulting chemical potentials.
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Affiliation(s)
- G Pastore
- Dipartimento di Fisica Teorica, Università di Trieste, Strada Costiera 11, I-34100 Trieste, Italy.
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26
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Duda Y, Vázquez F. Modeling of composite latex particle morphology by off-lattice Monte Carlo simulation. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2005; 21:1096-1102. [PMID: 15667196 DOI: 10.1021/la048065t] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Composite latex particles have shown a great range of applications such as paint resins, varnishes, water borne adhesives, impact modifiers, etc. The high-performance properties of this kind of materials may be explained in terms of a synergistical combination of two different polymers (usually a rubber and a thermoplastic). A great variety of composite latex particles with very different morphologies may be obtained by two-step emulsion polymerization processes. The formation of specific particle morphology depends on the chemical and physical nature of the monomers used during the synthesis, the process temperature, the reaction initiator, the surfactants, etc. Only a few models have been proposed to explain the appearance of the composite particle morphologies. These models have been based on the change of the interfacial energies during the synthesis. In this work, we present a new three-component model: Polymer blend (flexible and rigid chain particles) is dispersed in water by forming spherical cavities. Monte Carlo simulations of the model in two dimensions are used to determine the density distribution of chains and water molecules inside the suspended particle. This approach allows us to study the dependence of the morphology of the composite latex particles on the relative hydrophilicity and flexibility of the chain molecules as well as on their density and composition. It has been shown that our simple model is capable of reproducing the main features of the various morphologies observed in synthesis experiments.
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Affiliation(s)
- Yurko Duda
- Programa de Ingeniería Molecular, Instituto Mexicano del Petróleo, 07700 México, D.F., México.
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27
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Santos A, López de Haro M, Yuste SB. Equation of state of nonadditive d-dimensional hard-sphere mixtures. J Chem Phys 2005; 122:024514. [PMID: 15638605 DOI: 10.1063/1.1832591] [Citation(s) in RCA: 37] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
An equation of state for a multicomponent mixture of nonadditive hard spheres in d dimensions is proposed. It yields a rather simple density dependence and constitutes a natural extension of the equation of state for additive hard spheres proposed by us [A. Santos, S. B. Yuste, and M. Lopez de Haro, Mol. Phys. 96, 1 (1999)]. The proposal relies on the known exact second and third virial coefficients and requires as input the compressibility factor of the one-component system. A comparison is carried out both with another recent theoretical proposal based on a similar philosophy and with the available exact results and simulation data in d=1, 2, and 3. Good general agreement with the reported values of the virial coefficients and of the compressibility factor of binary mixtures is observed, especially for high asymmetries and/or positive nonadditivities.
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Affiliation(s)
- A Santos
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain.
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28
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Jagannathan K, Reddy G, Yethiraj A. Integral Equation Theory for Symmetric Nonadditive Hard Sphere Mixtures. J Phys Chem B 2004; 109:6764-8. [PMID: 16851761 DOI: 10.1021/jp045515h] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
An integral equation theory is presented for the pair correlation functions and phase behavior of symmetric nonadditive hard sphere mixtures with hard sphere diameters given by sigma(A)(A)() = sigma(BB) = lambdad and sigma(AB) = d. This mixture exhibits a fluid-fluid phase separation into an A-rich phase and a B-rich phase at high densities. The theory incorporates, into the closure approximation, all terms that can be calculated exactly in the density expansion of the direct correlation functions. We find that the closure approximation developed in this work is accurate for the structure and phase behavior over the entire range of lambda, when compared to computer simulations, and is significantly more accurate than the previous theories.
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Affiliation(s)
- Kamakshi Jagannathan
- Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, USA
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29
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Duda Y, Pizio O, Sokolowski S. Nonadditive Binary Hard Sphere Mixture in Disordered Hard Sphere Matrices: Integral Equations and Computer Simulation. J Phys Chem B 2004. [DOI: 10.1021/jp040340x] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Y. Duda
- Instituto Mexicano de Petroleo, Mexico D.F., Mexico, Institute of Condensed Matter Physics, National Academy of Sciences, Lviv, Ukraine, Instituto de Quimica de la UNAM, Coyoacan 04510, Mexico, D.F., and Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska University, Lublin, Poland
| | - O. Pizio
- Instituto Mexicano de Petroleo, Mexico D.F., Mexico, Institute of Condensed Matter Physics, National Academy of Sciences, Lviv, Ukraine, Instituto de Quimica de la UNAM, Coyoacan 04510, Mexico, D.F., and Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska University, Lublin, Poland
| | - S. Sokolowski
- Instituto Mexicano de Petroleo, Mexico D.F., Mexico, Institute of Condensed Matter Physics, National Academy of Sciences, Lviv, Ukraine, Instituto de Quimica de la UNAM, Coyoacan 04510, Mexico, D.F., and Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska University, Lublin, Poland
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30
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Bomont JM, Bretonnet JL. A consistent integral equation theory for hard spheres. J Chem Phys 2004; 121:1548-52. [PMID: 15260701 DOI: 10.1063/1.1764772] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The standard integral equation approach is used to extract the bridge function and other correlation functions of hard spheres fluid. To achieve this, we first use a recent consistent closure relation proposed by Bomont et al. [J. Chem. Phys. 119, 2188 (2003)] that has already proven to be accurate to describe the Lennard-Jones fluid properties. Second, we take advantage of the coherent scheme derived by Bomont [J. Chem. Phys. 119, 11484 (2003)] to calculate the excess chemical potential, the entropy and some relative transport properties. Very good agreement is obtained for structural quantities and thermodynamic properties as compared to exact data at densities ranging from 0.1 to 0.9.
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Affiliation(s)
- Jean-Marc Bomont
- Equipe de Chimie et Biochimie Theoriques, UMR CNRS-UHP 7565, Universite Henri Poincare Nancy I, F-54509 Vandoeuvre-Les-Nancy, France.
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31
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Pellicane G, Caccamo C, Wilson DS, Lee LL. Replica Ornstein-Zernike self-consistent theory for mixtures in random pores. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:061202. [PMID: 15244549 DOI: 10.1103/physreve.69.061202] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2003] [Indexed: 05/24/2023]
Abstract
We present a self-consistent integral equation theory for a binary liquid in equilibrium with a disordered medium, based on the formalism of the replica Ornstein-Zernike (ROZ) equations. Specifically, we derive direct formulas for the chemical potentials and the zero-separation theorems (the latter provide a connection between the chemical potentials and the fluid cavity distribution functions). Next we solve a modified-Verlet closure to ROZ equations, which has built-in parameters that can be adjusted to satisfy the zero-separation theorems. The degree of thermodynamic consistency of the theory is also kept under control. We model the binary fluid in random pores as a symmetrical binary mixture of nonadditive hard spheres in a disordered hard-sphere matrix and consider two different values of the nonadditivity parameter and of the quenched matrix packing fraction, at different mixture concentrations. We compare the theoretical structural properties as obtained through the present approach with Percus-Yevick and Martinov-Sarkisov integral equation theories, and assess both structural and thermodynamic properties by performing canonical standard and biased grand canonical Monte Carlo simulations. Our theory appears superior to the other integral equation schemes here examined and provides reliable estimates of the chemical potentials. This feature should be useful in studying the fluid phase behavior of model adsorbates in random pores in general.
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Affiliation(s)
- G Pellicane
- Dipartimento di Fisica, Università di Messina and Istituto Nazionale per la Fisica della Materia (INFM), Messina, Italy
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32
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33
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34
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Viravathana P, Marr DWM. Synthesis of colloidal aluminosilicate for light-scattering investigations. J Colloid Interface Sci 2003; 265:15-22. [PMID: 12927158 DOI: 10.1016/s0021-9797(03)00369-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Abstract
To develop a binary colloidal system with a slight index of refraction mismatch suitable for light scattering studies, pure silica particles synthesized by the method of Stöber were mixed with aluminosilicate colloids synthesized using a novel approach. With this, index-matching for one component allowed extraction of the spatial distribution of the other. In addition, it was observed that by varying the solvent, interactions between colloids could be tuned from purely repulsive to weakly attractive.
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Affiliation(s)
- P Viravathana
- Chemistry Department, Kasetsart University, Bangkok 10900, Thailand.
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35
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Góźdź WT. Critical-point and coexistence curve properties of a symmetric mixture of nonadditive hard spheres: A finite size scaling study. J Chem Phys 2003. [DOI: 10.1063/1.1589746] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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36
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Jagannathan K, Yethiraj A. Monte Carlo simulations for the phase behavior of symmetric nonadditive hard sphere mixtures. J Chem Phys 2003. [DOI: 10.1063/1.1563595] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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37
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Stability and interfacial properties of confined nonadditive hard-sphere binary mixture. J Colloid Interface Sci 2003. [DOI: 10.1016/s0021-9797(02)00042-5] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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38
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Saija F, Giaquinta PV. Entropy and Fluid−Fluid Separation in Nonadditive Hard-Sphere Mixtures: The Asymmetric Case. J Phys Chem B 2002. [DOI: 10.1021/jp013150k] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- F. Saija
- Istituto di Tecniche Spettroscopiche del CNR, Via La Farina 237, 98123, Messina, Italy, and Istituto Nazionale per la Fisica della Materia, Unità di Ricerca di Messina, and Dipartimento di Fisica, Università degli Studi di Messina, Contrada Papardo, 98166 Messina, Italy
| | - P. V. Giaquinta
- Istituto di Tecniche Spettroscopiche del CNR, Via La Farina 237, 98123, Messina, Italy, and Istituto Nazionale per la Fisica della Materia, Unità di Ricerca di Messina, and Dipartimento di Fisica, Università degli Studi di Messina, Contrada Papardo, 98166 Messina, Italy
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39
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Jorge S, Lomba E, Abascal JLF. Theory and simulation of the triplet structure factor and triplet direct correlation functions in binary mixtures. J Chem Phys 2002. [DOI: 10.1063/1.1426418] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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40
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Fernaud MJ, Lomba E, Weis JJ. Adsorption of a diatomic molecular fluid into random porous media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:051501. [PMID: 11735923 DOI: 10.1103/physreve.64.051501] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2001] [Indexed: 05/23/2023]
Abstract
Structural and thermodynamic properties of a homonuclear hard dumbbell fluid adsorbed into a disordered hard sphere matrix are studied by means of integral equation techniques and computer simulation. In particular, we have rewritten the replica Ornstein-Zernike equations to deal with orientational degrees of freedom and we have solved them in two different approaches: the hypernetted chain equation and a semiempirical extension of Verlet's approximation. We have also derived direct expressions to calculate the chemical potential in these approximations. Comparison with grand canonical Monte Carlo results shows that both theoretical treatments describe adequately the physical behavior of the system, Verlet's approach being, however, clearly superior in accordance with previous findings for equilibrated hard core mixtures.
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Affiliation(s)
- M J Fernaud
- Instituto de Química Física Rocasolano, CSIC, Serrano 119, E-28006 Madrid, Spain
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41
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Almarza NG, Enciso E, García MF, González MA, Bermejo FJ. Reentrant miscibility in fluids with spherical interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:012501. [PMID: 11461308 DOI: 10.1103/physreve.64.012501] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2001] [Indexed: 05/23/2023]
Abstract
We have obtained the closed-loop fluid-fluid immiscibility in the phase diagram of a binary mixture with interactions with spherical symmetry. That topology appears when a short-range attractive interaction is considered between unlike pair molecules. We present "exact" results obtained from Monte Carlo simulation on different ensembles and results from the application of a first-order perturbation theory.
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Affiliation(s)
- N G Almarza
- Departamento de Química Física I, Facultad de Ciencias Químicas, Universidad Complutense, 28040 Madrid, Spain
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42
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Lee LL, Malijevsky A. Structures and properties of hard sphere mixtures based on a self-consistent integral equation. J Chem Phys 2001. [DOI: 10.1063/1.1359182] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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43
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Jorge S, Kahl G, Lomba E, Abascal JLF. On the triplet structure of binary liquids. J Chem Phys 2000. [DOI: 10.1063/1.1287337] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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44
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Fernaud MJ, Lomba E, Lee LL. Chemical potentials and potential distributions of inclusion gas in quenched-annealed random porous media. J Chem Phys 1999. [DOI: 10.1063/1.480376] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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45
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Saija F, Pastore G, Giaquinta PV. Entropy and Fluid−Fluid Separation in Nonadditive Hard-Sphere Mixtures. J Phys Chem B 1998. [DOI: 10.1021/jp982202b] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- F. Saija
- Istituto Tecniche Spettroscopiche, CNR, Contrada Papardo, Salita Sperone 31, 98166 Messina, Italy, Istituto Nazionale per la Fisica della Materia, Unità di Ricerca di Trieste, Strada Costiera 11, 34014 Trieste, Italy, Università degli Studi di Trieste, Dipartimento di Fisica Teorica, Strada Costiera 11, 34014 Trieste, Italy, Istituto Nazionale per la Fisica della Materia, Unità di Ricerca di Messina, Contrada Papardo, 98166 Messina, Italy, and Università degli Studi di Messina, Dipartimento di Fisica,
| | - G. Pastore
- Istituto Tecniche Spettroscopiche, CNR, Contrada Papardo, Salita Sperone 31, 98166 Messina, Italy, Istituto Nazionale per la Fisica della Materia, Unità di Ricerca di Trieste, Strada Costiera 11, 34014 Trieste, Italy, Università degli Studi di Trieste, Dipartimento di Fisica Teorica, Strada Costiera 11, 34014 Trieste, Italy, Istituto Nazionale per la Fisica della Materia, Unità di Ricerca di Messina, Contrada Papardo, 98166 Messina, Italy, and Università degli Studi di Messina, Dipartimento di Fisica,
| | - P. V. Giaquinta
- Istituto Tecniche Spettroscopiche, CNR, Contrada Papardo, Salita Sperone 31, 98166 Messina, Italy, Istituto Nazionale per la Fisica della Materia, Unità di Ricerca di Trieste, Strada Costiera 11, 34014 Trieste, Italy, Università degli Studi di Trieste, Dipartimento di Fisica Teorica, Strada Costiera 11, 34014 Trieste, Italy, Istituto Nazionale per la Fisica della Materia, Unità di Ricerca di Messina, Contrada Papardo, 98166 Messina, Italy, and Università degli Studi di Messina, Dipartimento di Fisica,
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46
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47
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Caccamo C, Costa D, Pellicane G. A comprehensive study of the phase diagram of symmetrical hard-core Yukawa mixtures. J Chem Phys 1998. [DOI: 10.1063/1.477053] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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48
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Saija F, Fiumara G, Giaquinta PV. Virial expansion of a non-additive hard-sphere mixture. J Chem Phys 1998. [DOI: 10.1063/1.476355] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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49
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Lee LL. The potential distribution-based closures to the integral equations for liquid structure: The Lennard-Jones fluid. J Chem Phys 1997. [DOI: 10.1063/1.474974] [Citation(s) in RCA: 39] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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50
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Sear RP, Mulder BM. Phase behaviour of a symmetric binary mixture of hard rods. J Chem Phys 1996. [DOI: 10.1063/1.473001] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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