Luis Macdowell - Academia.edu (original) (raw)

Papers by Luis Macdowell

En esta memoria se estudian las propiedades críticas y el equilibrio líquido-vapor de moléculas f... more En esta memoria se estudian las propiedades críticas y el equilibrio líquido-vapor de moléculas fléxibles, utilizando para ello las teorías de perturbaciones, las teorías de ecuaciones integrales(MSA y RHNC) y los métodos de simulación molecular de Monte Carlo. Se propone una teoría de perturbaciones de tipo Van der Waals, en la que el sistema de referencia se describe mediante una modificación de la ecuación de Wertheim, y la contribución perturbativa se determina mediante un término de campo medio evaluado exactamente. La teoría se aplica a un modelo realista de alcano, en el que los distintos átomos de carbono se tratan explícitamente. De este modo, es posible describir cualitativamente las propiedades más significativas de alcanos lineales y ramificados, tanto en estado puro como en mezclas binarias. Así mismo, se predice el comportamiento crítico en el límite de tamaños infinitos, empleando para ello tanto la teoría de perturbaciones de campo medio, como la teoría de asociación...

Coarse-Graining of Condensed Phase and Biomolecular Systems, 2008

Physical Review E, 2007

We develop a methodology for the calculation of surface free energies based on the probability di... more We develop a methodology for the calculation of surface free energies based on the probability distribution of a wandering interface. Using a simple extension of the NpT sampling, we allow the interface area to randomly probe the available space and evaluate the surface free energy from histogram analysis and the corresponding average. The method is suitable for studying systems with either continuous or discontinuous potentials, as it does not require explicit evaluation of the virial. The proposed algorithm is compared with known results for the surface tension of Lennard-Jones and Square Well fluid, as well as for the interface tension of a bead-spring polymer model and good agreement is found. We also calculate interfacial tensions of freely jointed tangent hard sphere chains on athermal walls for a wide range of chain lengths and densities. The results are compared with three different theoretical approaches, Scaled Particle Theory, the Yu and Wu density functional theory and an analytical approximation based on the latter approach. Whereas SPT only yields qualitative results, the last two approaches are found to yield very good agreement with simulations.

Molecular Physics, 1999

ABSTRACT

The Journal of Chemical Physics, 2006

Computer simulations are employed to obtain subcritical isotherms of small finite sized systems i... more Computer simulations are employed to obtain subcritical isotherms of small finite sized systems inside the coexistence region. For all temperatures considered, ranging from the triple point up to the critical point, the isotherms gradually developed a sequence of sharp discontinuities as the system size increased from ϳ8 to ϳ21 molecular diameters. For the smallest system sizes, and more so close to the critical point, the isotherms appeared smooth, resembling the continuous van der Waals loop obtained from extrapolation of an analytic equation of state outside the coexistence region. As the system size was increased, isotherms in the chemical potential-density plane developed first two, then four, and finally six discontinuities. Visual inspection of selected snapshots revealed that the observed discontinuities are related to structural transitions between droplets ͑on the vapor side͒ and bubbles ͑on the liquid side͒ of spherical, cylindrical, and tetragonal shapes. A capillary drop model was developed to qualitatively rationalize these observations. Analytic results were obtained and found to be in full agreement with the computer simulation results. The analysis shows that the shape of the subcritical isotherms is dictated by a single characteristic volume ͑or length scale͒, which depends on the surface tension, compressibility, and coexistence densities. For small reduced system volumes, the model predicts that a homogeneous fluid is stable across the whole coexistence region, thus explaining the continuous van der Waals isotherms observed in the simulations. When the liquid and vapor free energies are described by means of an accurate mean-field equation of state and surface tensions from simulation are employed, the capillary model is found to describe the simulated isotherms accurately, especially for large systems ͑i.e., larger than about 15 molecular diameters͒ at low temperature ͑lower than about 0.85 times the critical temperature͒. This implies that the Laplace pressure differences can be predicted for drops as small as five molecular diameters, and as few as about 500 molecules. The theoretical study also shows that the extrema or apparent spinodal points of the finite size loops are more closely related to ͑finite system size͒ bubble and dew points than to classical spinodals. Our results are of relevance to phase transitions in nanopores and show that first order corrections to nucleation energies in finite closed systems are power laws of the inverse volume.

The Journal of Chemical Physics, 2003

... this observation caused Schulman to disregard his own work on metastability as relevant ... t... more ... this observation caused Schulman to disregard his own work on metastability as relevant ... the attainable supersaturation is limited by the system size, because for large volumes the ... V. Talanquer, “A new phenomenological approach to gas--liquid nucleation based on the scaling ...

The Journal of Chemical Physics, 2009

The prediction of the equation of state and the phase behavior of simple fluids (noble gases, car... more The prediction of the equation of state and the phase behavior of simple fluids (noble gases, carbon dioxide, benzene, methane, short alkane chains) and their mixtures by Monte Carlo computer simulation and analytic approximations based on thermodynamic perturbation theory is discussed. Molecules are described by coarse grained (CG) models, where either the whole molecule (carbon dioxide, benzene, methane) or a group of a few successive CH 2 groups (in the case of alkanes) are lumped into an effective point particle. Interactions among these point particles are fitted by Lennard-Jones (LJ) potentials such that the vapor-liquid critical point of the fluid is reproduced in agreement with experiment; in the case of quadrupolar molecules a quadrupole-quadrupole interaction is included. These models are shown to provide a satisfactory description of the liquid-vapour phase diagram of these pure fluids. Investigations of mixtures, using the Lorentz-Berthelot (LB) combining rule, also produce satisfactory results if compared with experiment, while in some previous attempts (in which polar solvents were modelled without explicitly taking into account quadrupolar interaction), strong violations of the LB rules were required. For this reason, the present investigation is a step towards predictive

The Journal of Chemical Physics, 2011

Surface instabilities of minority chains in dense polymer brushes: A comparison of density functi... more Surface instabilities of minority chains in dense polymer brushes: A comparison of density functional theory and quasi-off-lattice self-consistent field theory Analysis of self-consistency effects in range-separated density-functional theory with Møller-Plesset perturbation theory Periodic density functional embedding theory for complete active space self-consistent field and configuration interaction calculations: Ground and excited states Using polyatomic density functional theory of Kierlik and Rosinberg, we show that Wertheim's thermodynamic perturbation theory (TPT) incorporates solvation effects in a systematic, although simplified form. We derive two approximate solvation potentials, which require the knowledge of the correlation function in the reference unbonded fluid only. The theoretical predictions are tested against many-chain Monte Carlo simulations for moderate chain lengths. The predictions of the end-to-end distance in the bulk are in a reasonable agreement with simulations for the TPT(M-1) approximation, while the simpler TPT2_e approximation leads to the solvation potential that is shorter ranged and considerably less accurate. The resulting conformations are used in the subsequent selfconsistent field theory calculations of hard-sphere polymers at a hard wall. While the incorporation of the solvation effects has little impact on the density profiles, the predictions of the components of the end-to-end distance vector as a function of the distance to the wall are much improved.

The Journal of Chemical Physics, 2008

We study the properties of athermal polymers at hard walls using two different versions of self-c... more We study the properties of athermal polymers at hard walls using two different versions of self-consistent field theory ͑SCFT͒. We calculate the segment density profiles, center of mass profiles, bond orientation vector profiles, and end-to-end vector distributions and compare with grand canonical Monte Carlo simulations. Using the same excess free energy prescription for both theories, we investigate the role of the excluded volume intramolecular interactions on these properties, show the relation between SCFT and density functional theory, and discuss several numerical implementations of the SCFT method. The phantom chain model gives Gaussian chain statistics independent of the conditions. Including the full intramolecular potential leads to an improved description of the low density regime but it does not produce any significant improvement in the semidiluted and concentrated regimes. We show that a viable compromise is achieved by using the effective field resulting from the phantom chain model and by calculating single chain properties using the full intramolecular potential. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp ps ͑b͒ = 0.4 3 , M = 16; and ͑d͒ ps ͑b͒ = 0.6 3 , M = 16. The circles denote GCMC data; the solid and the dashed lines denote SCFT and DFT results, respectively. The red dotted lines are the results of the mixed method ͑see text͒.

The European Physical Journal Special Topics, 2011

Comments are provided about the difficulty of measuring the solid-fluid surface tension as emphas... more Comments are provided about the difficulty of measuring the solid-fluid surface tension as emphasized by Binder et al. [9], Marmur [8] and Sefiane [7].

The European Physical Journal Special Topics, 2011

We discuss the feasibility of a hierarchical protocol whereby the description and prediction of a... more We discuss the feasibility of a hierarchical protocol whereby the description and prediction of adsorbed fluids in confined systems at the mesoscopic scale is achieved by use of interface potentials that are obtained from raw molecular simulation data. Starting from a microscopic description of a fluid's interface on a flat substrate, we attempt to extract the minimum information that is required in order to predict the behavior of that fluid at larger length scales from coarse grained surface Hamiltonians. A critical assessment of this procedure hinges on controversial aspects of wetting behavior and more generally on the meaning of metastability and instability of thermodynamic systems. a

Molecular Physics, 2015

ABSTRACT A new equation of state is found to reproduce analytically the thermodynamic properties ... more ABSTRACT A new equation of state is found to reproduce analytically the thermodynamic properties of fluids interacting with a spherical Kihara potential formulated for arbitrary values of the inner hard core. The results from the expressions developed are satisfactorily compared with the Monte Carlo simulation data for a wide range of densities and temperatures. The foundation of the new equation is strictly theoretical, based on the first-order mean spherical approximation perturbation approach of Tang and Lu, and provides a good starting point for the description of polyatomic fluids within the benchmark of Wertheim's first-order perturbation theory of associating fluids.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2015

We use Monte Carlo simulations of a Lennard-Jones fluid adsorbed on a short-range planar wall sub... more We use Monte Carlo simulations of a Lennard-Jones fluid adsorbed on a short-range planar wall substrate to study the fluctuations in the thickness of the wetting layer, and we get a quantitative and consistent characterization of their mesoscopic Hamiltonian, H[ξ]. We have observed important finite-size effects, which were hampering the analysis of previous results obtained with smaller systems. The results presented here support an appealing simple functional form for H[ξ], close but not exactly equal to the theoretical nonlocal proposal made on the basis a generic density-functional analysis by Parry and coworkers. We have analyzed systems under different wetting conditions, as a proof of principle for a method that provides a quantitative bridge between the molecular interactions and the phenomenology of wetting films at mesoscopic scales.

We report numerical results for the third virial coefficient of two center Lennard-Jones quadrupo... more We report numerical results for the third virial coefficient of two center Lennard-Jones quadrupolar molecules. Calculations are performed for 35 models with different elongations and quadrupoles over a temperature range from half to twice the critical temperature. It is found that increasing the elongation at fixed quadrupole has the effect of increasing B 3 . On the other hand, at fixed elongation B 3 first decreases with increasing quadrupole at low temperatures, then increases with increasing quadrupole at higher temperatures. We estimate the temperature at which the third virial coefficient vanishes. Although both this temperature and the critical temperature increase with the quadrupole moment, their ratio remains almost constant. We predict the critical properties using two different truncated virial series. The first one employs the exact second and third virial coefficients. The second one approximates the fourth order contribution by using estimates obtained for hard diatomics. It is found that both methods yield fairly good predictions, with a somewhat better performance of the approximate fourth order expansion. The two methods are complementary, however, because they consistently bracket the exact value as determined from computer simulations.

Springer Proceedings in Physics, 2004

A fluid in the N V T ensemble at T less than the critical temperature T c and ρ = N/V somewhat in... more A fluid in the N V T ensemble at T less than the critical temperature T c and ρ = N/V somewhat in excess of ρ (1) coex (density of the saturated gas in the gas-liquid transition) is considered. For V → ∞, a macroscopic liquid droplet coexists with surrounding saturated gas according to the lever rule. For finite V , droplets can only exist if they exceed a minimum size. A (rounded) first order transition of the system occurs when the droplet evaporates into the supersaturated gas. Simulation evidence for this transition is given for a Lennard-Jones model and interpreted by a phenomenological theory. At the transition, the chemical potential difference µ t − µ coex scales like L −d/(d+1) for a cubic volume V = L d in d dimensions, as L → ∞.

Physica B Condensed Matter

... ELSEVIER Physica B 234 236 (1997) 106-108 Rotational molecular dynamics in the phase of n-non... more ... ELSEVIER Physica B 234 236 (1997) 106-108 Rotational molecular dynamics in the phase of n-nonadecane L. Gonzalez Mac Dowella, F. Guillaumeb'*, J.-P. Ryckaerta, P. Girardb,c, V. Rodriguezb, A.-J. Dianouxc aPhysique statistique de la Mati~re Condens~e, CP223, ULB ...

En esta memoria se estudian las propiedades críticas y el equilibrio líquido-vapor de moléculas f... more En esta memoria se estudian las propiedades críticas y el equilibrio líquido-vapor de moléculas fléxibles, utilizando para ello las teorías de perturbaciones, las teorías de ecuaciones integrales(MSA y RHNC) y los métodos de simulación molecular de Monte Carlo. Se propone una teoría de perturbaciones de tipo Van der Waals, en la que el sistema de referencia se describe mediante una modificación de la ecuación de Wertheim, y la contribución perturbativa se determina mediante un término de campo medio evaluado exactamente. La teoría se aplica a un modelo realista de alcano, en el que los distintos átomos de carbono se tratan explícitamente. De este modo, es posible describir cualitativamente las propiedades más significativas de alcanos lineales y ramificados, tanto en estado puro como en mezclas binarias. Así mismo, se predice el comportamiento crítico en el límite de tamaños infinitos, empleando para ello tanto la teoría de perturbaciones de campo medio, como la teoría de asociación...

Coarse-Graining of Condensed Phase and Biomolecular Systems, 2008

Physical Review E, 2007

We develop a methodology for the calculation of surface free energies based on the probability di... more We develop a methodology for the calculation of surface free energies based on the probability distribution of a wandering interface. Using a simple extension of the NpT sampling, we allow the interface area to randomly probe the available space and evaluate the surface free energy from histogram analysis and the corresponding average. The method is suitable for studying systems with either continuous or discontinuous potentials, as it does not require explicit evaluation of the virial. The proposed algorithm is compared with known results for the surface tension of Lennard-Jones and Square Well fluid, as well as for the interface tension of a bead-spring polymer model and good agreement is found. We also calculate interfacial tensions of freely jointed tangent hard sphere chains on athermal walls for a wide range of chain lengths and densities. The results are compared with three different theoretical approaches, Scaled Particle Theory, the Yu and Wu density functional theory and an analytical approximation based on the latter approach. Whereas SPT only yields qualitative results, the last two approaches are found to yield very good agreement with simulations.

Molecular Physics, 1999

ABSTRACT

The Journal of Chemical Physics, 2006

Computer simulations are employed to obtain subcritical isotherms of small finite sized systems i... more Computer simulations are employed to obtain subcritical isotherms of small finite sized systems inside the coexistence region. For all temperatures considered, ranging from the triple point up to the critical point, the isotherms gradually developed a sequence of sharp discontinuities as the system size increased from ϳ8 to ϳ21 molecular diameters. For the smallest system sizes, and more so close to the critical point, the isotherms appeared smooth, resembling the continuous van der Waals loop obtained from extrapolation of an analytic equation of state outside the coexistence region. As the system size was increased, isotherms in the chemical potential-density plane developed first two, then four, and finally six discontinuities. Visual inspection of selected snapshots revealed that the observed discontinuities are related to structural transitions between droplets ͑on the vapor side͒ and bubbles ͑on the liquid side͒ of spherical, cylindrical, and tetragonal shapes. A capillary drop model was developed to qualitatively rationalize these observations. Analytic results were obtained and found to be in full agreement with the computer simulation results. The analysis shows that the shape of the subcritical isotherms is dictated by a single characteristic volume ͑or length scale͒, which depends on the surface tension, compressibility, and coexistence densities. For small reduced system volumes, the model predicts that a homogeneous fluid is stable across the whole coexistence region, thus explaining the continuous van der Waals isotherms observed in the simulations. When the liquid and vapor free energies are described by means of an accurate mean-field equation of state and surface tensions from simulation are employed, the capillary model is found to describe the simulated isotherms accurately, especially for large systems ͑i.e., larger than about 15 molecular diameters͒ at low temperature ͑lower than about 0.85 times the critical temperature͒. This implies that the Laplace pressure differences can be predicted for drops as small as five molecular diameters, and as few as about 500 molecules. The theoretical study also shows that the extrema or apparent spinodal points of the finite size loops are more closely related to ͑finite system size͒ bubble and dew points than to classical spinodals. Our results are of relevance to phase transitions in nanopores and show that first order corrections to nucleation energies in finite closed systems are power laws of the inverse volume.

The Journal of Chemical Physics, 2003

... this observation caused Schulman to disregard his own work on metastability as relevant ... t... more ... this observation caused Schulman to disregard his own work on metastability as relevant ... the attainable supersaturation is limited by the system size, because for large volumes the ... V. Talanquer, “A new phenomenological approach to gas--liquid nucleation based on the scaling ...

The Journal of Chemical Physics, 2009

The prediction of the equation of state and the phase behavior of simple fluids (noble gases, car... more The prediction of the equation of state and the phase behavior of simple fluids (noble gases, carbon dioxide, benzene, methane, short alkane chains) and their mixtures by Monte Carlo computer simulation and analytic approximations based on thermodynamic perturbation theory is discussed. Molecules are described by coarse grained (CG) models, where either the whole molecule (carbon dioxide, benzene, methane) or a group of a few successive CH 2 groups (in the case of alkanes) are lumped into an effective point particle. Interactions among these point particles are fitted by Lennard-Jones (LJ) potentials such that the vapor-liquid critical point of the fluid is reproduced in agreement with experiment; in the case of quadrupolar molecules a quadrupole-quadrupole interaction is included. These models are shown to provide a satisfactory description of the liquid-vapour phase diagram of these pure fluids. Investigations of mixtures, using the Lorentz-Berthelot (LB) combining rule, also produce satisfactory results if compared with experiment, while in some previous attempts (in which polar solvents were modelled without explicitly taking into account quadrupolar interaction), strong violations of the LB rules were required. For this reason, the present investigation is a step towards predictive

The Journal of Chemical Physics, 2011

Surface instabilities of minority chains in dense polymer brushes: A comparison of density functi... more Surface instabilities of minority chains in dense polymer brushes: A comparison of density functional theory and quasi-off-lattice self-consistent field theory Analysis of self-consistency effects in range-separated density-functional theory with Møller-Plesset perturbation theory Periodic density functional embedding theory for complete active space self-consistent field and configuration interaction calculations: Ground and excited states Using polyatomic density functional theory of Kierlik and Rosinberg, we show that Wertheim's thermodynamic perturbation theory (TPT) incorporates solvation effects in a systematic, although simplified form. We derive two approximate solvation potentials, which require the knowledge of the correlation function in the reference unbonded fluid only. The theoretical predictions are tested against many-chain Monte Carlo simulations for moderate chain lengths. The predictions of the end-to-end distance in the bulk are in a reasonable agreement with simulations for the TPT(M-1) approximation, while the simpler TPT2_e approximation leads to the solvation potential that is shorter ranged and considerably less accurate. The resulting conformations are used in the subsequent selfconsistent field theory calculations of hard-sphere polymers at a hard wall. While the incorporation of the solvation effects has little impact on the density profiles, the predictions of the components of the end-to-end distance vector as a function of the distance to the wall are much improved.

The Journal of Chemical Physics, 2008

We study the properties of athermal polymers at hard walls using two different versions of self-c... more We study the properties of athermal polymers at hard walls using two different versions of self-consistent field theory ͑SCFT͒. We calculate the segment density profiles, center of mass profiles, bond orientation vector profiles, and end-to-end vector distributions and compare with grand canonical Monte Carlo simulations. Using the same excess free energy prescription for both theories, we investigate the role of the excluded volume intramolecular interactions on these properties, show the relation between SCFT and density functional theory, and discuss several numerical implementations of the SCFT method. The phantom chain model gives Gaussian chain statistics independent of the conditions. Including the full intramolecular potential leads to an improved description of the low density regime but it does not produce any significant improvement in the semidiluted and concentrated regimes. We show that a viable compromise is achieved by using the effective field resulting from the phantom chain model and by calculating single chain properties using the full intramolecular potential. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp ps ͑b͒ = 0.4 3 , M = 16; and ͑d͒ ps ͑b͒ = 0.6 3 , M = 16. The circles denote GCMC data; the solid and the dashed lines denote SCFT and DFT results, respectively. The red dotted lines are the results of the mixed method ͑see text͒.

The European Physical Journal Special Topics, 2011

Comments are provided about the difficulty of measuring the solid-fluid surface tension as emphas... more Comments are provided about the difficulty of measuring the solid-fluid surface tension as emphasized by Binder et al. [9], Marmur [8] and Sefiane [7].

The European Physical Journal Special Topics, 2011

We discuss the feasibility of a hierarchical protocol whereby the description and prediction of a... more We discuss the feasibility of a hierarchical protocol whereby the description and prediction of adsorbed fluids in confined systems at the mesoscopic scale is achieved by use of interface potentials that are obtained from raw molecular simulation data. Starting from a microscopic description of a fluid's interface on a flat substrate, we attempt to extract the minimum information that is required in order to predict the behavior of that fluid at larger length scales from coarse grained surface Hamiltonians. A critical assessment of this procedure hinges on controversial aspects of wetting behavior and more generally on the meaning of metastability and instability of thermodynamic systems. a

Molecular Physics, 2015

ABSTRACT A new equation of state is found to reproduce analytically the thermodynamic properties ... more ABSTRACT A new equation of state is found to reproduce analytically the thermodynamic properties of fluids interacting with a spherical Kihara potential formulated for arbitrary values of the inner hard core. The results from the expressions developed are satisfactorily compared with the Monte Carlo simulation data for a wide range of densities and temperatures. The foundation of the new equation is strictly theoretical, based on the first-order mean spherical approximation perturbation approach of Tang and Lu, and provides a good starting point for the description of polyatomic fluids within the benchmark of Wertheim's first-order perturbation theory of associating fluids.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2015

We use Monte Carlo simulations of a Lennard-Jones fluid adsorbed on a short-range planar wall sub... more We use Monte Carlo simulations of a Lennard-Jones fluid adsorbed on a short-range planar wall substrate to study the fluctuations in the thickness of the wetting layer, and we get a quantitative and consistent characterization of their mesoscopic Hamiltonian, H[ξ]. We have observed important finite-size effects, which were hampering the analysis of previous results obtained with smaller systems. The results presented here support an appealing simple functional form for H[ξ], close but not exactly equal to the theoretical nonlocal proposal made on the basis a generic density-functional analysis by Parry and coworkers. We have analyzed systems under different wetting conditions, as a proof of principle for a method that provides a quantitative bridge between the molecular interactions and the phenomenology of wetting films at mesoscopic scales.

We report numerical results for the third virial coefficient of two center Lennard-Jones quadrupo... more We report numerical results for the third virial coefficient of two center Lennard-Jones quadrupolar molecules. Calculations are performed for 35 models with different elongations and quadrupoles over a temperature range from half to twice the critical temperature. It is found that increasing the elongation at fixed quadrupole has the effect of increasing B 3 . On the other hand, at fixed elongation B 3 first decreases with increasing quadrupole at low temperatures, then increases with increasing quadrupole at higher temperatures. We estimate the temperature at which the third virial coefficient vanishes. Although both this temperature and the critical temperature increase with the quadrupole moment, their ratio remains almost constant. We predict the critical properties using two different truncated virial series. The first one employs the exact second and third virial coefficients. The second one approximates the fourth order contribution by using estimates obtained for hard diatomics. It is found that both methods yield fairly good predictions, with a somewhat better performance of the approximate fourth order expansion. The two methods are complementary, however, because they consistently bracket the exact value as determined from computer simulations.

Springer Proceedings in Physics, 2004

A fluid in the N V T ensemble at T less than the critical temperature T c and ρ = N/V somewhat in... more A fluid in the N V T ensemble at T less than the critical temperature T c and ρ = N/V somewhat in excess of ρ (1) coex (density of the saturated gas in the gas-liquid transition) is considered. For V → ∞, a macroscopic liquid droplet coexists with surrounding saturated gas according to the lever rule. For finite V , droplets can only exist if they exceed a minimum size. A (rounded) first order transition of the system occurs when the droplet evaporates into the supersaturated gas. Simulation evidence for this transition is given for a Lennard-Jones model and interpreted by a phenomenological theory. At the transition, the chemical potential difference µ t − µ coex scales like L −d/(d+1) for a cubic volume V = L d in d dimensions, as L → ∞.

Physica B Condensed Matter

... ELSEVIER Physica B 234 236 (1997) 106-108 Rotational molecular dynamics in the phase of n-non... more ... ELSEVIER Physica B 234 236 (1997) 106-108 Rotational molecular dynamics in the phase of n-nonadecane L. Gonzalez Mac Dowella, F. Guillaumeb'*, J.-P. Ryckaerta, P. Girardb,c, V. Rodriguezb, A.-J. Dianouxc aPhysique statistique de la Mati~re Condens~e, CP223, ULB ...