Third Virial Coefficient for 4‐Arm and 6‐Arm Star Polymers

Caracciolo, Sergio; Mognetti, Bortolo Matteo; Pelissetto, Andrea

doi:10.1002/mats.200800001

Cited by 15 publications

(16 citation statements)

References 34 publications

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“…The formula for B 3 differs from that in the textbook case of zero field; the case of a nonzero field does not appear to have been covered in any standard reference. It turns out that the correct expression for B 3 contains an extra term which has also been found in the third virial coefficient for flexible molecules: the so-called "uniquely flexible" contribution [32,33,58]. The common feature appears to be the presence of intramolecular degrees of freedom contributing to the potential energy, which in the present case corresponds to the dipolar orientation coupling to the applied field.…”

Section: A Virial Coefficients For a System In An Applied Fieldmentioning

confidence: 69%

“…The correct expressions for these coefficients are derived quite generally for a one-component system in an applied field: the formula for B 2 is the same as in the zero-field case; the formula for B 3 , however, is different in an applied field. This is a surprising result which is not emphasized in standard texts, but which has been noticed before in the virial expansion for flexible molecules [32,33]. The analytical evaluation of the virial coefficients is based on an expansion in λ and a representation in terms of appropriate functions of α.…”

Section: Introductionmentioning

confidence: 81%

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Thermodynamics of ferrofluids in applied magnetic fields

2013

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The thermodynamic properties of ferrofluids in applied magnetic fields are examined using theory and computer simulation. The dipolar hard sphere model is used. The second and third virial coefficients (B 2 and B 3 ) are evaluated as functions of the dipolar coupling constant λ, and the Langevin parameter α. The formula for B 3 for a system in an applied field is different from that in the zero-field case, and a derivation is presented. The formulas are compared to results from Mayer-sampling calculations, and the trends with increasing λ and α are examined. Very good agreement between theory and computation is demonstrated for the realistic values λ 2. The analytical formulas for the virial coefficients are incorporated in to various forms of virial expansion, designed to minimize the effects of truncation. The theoretical results for the equation of state are compared against results from Monte Carlo simulations. In all cases, the so-called logarithmic free energy theory is seen to be superior. In this theory, the virial expansion of the Helmholtz free energy is re-summed in to a logarithmic function. Its success is due to the approximate representation of high-order terms in the virial expansion, while retaining the exact low-concentration behavior. The theory also yields the magnetization, and a comparison with simulation results and a competing modified mean-field theory shows excellent agreement. Finally, the putative field-dependent critical parameters for the condensation transition are obtained and compared against existing simulation results for the Stockmayer fluid. Dipolar hard spheres do not undergo the transition, but the presence of isotropic attractions, as in the Stockmayer fluid, gives rise to condensation even in zero field. A comparison of the relative changes in critical parameters with increasing field strength shows excellent agreement between theory and simulation, showing that the theoretical treatment of the dipolar interactions is robust.

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Section: A Virial Coefficients For a System In An Applied Fieldmentioning

confidence: 69%

Section: Introductionmentioning

confidence: 81%

Thermodynamics of ferrofluids in applied magnetic fields

2013

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“…Within the context of the virial equation of state for dilute gases, u ij is the interaction energy between molecules i and j. For osmotic virial coefficients, the context considered by Caracciolo et al, 2,3 u ij is the potential of mean force between particles i and j within a solvent. a) Author to whom correspondence should be addressed.…”

Section: Introductionmentioning

confidence: 99%

Mayer-sampling Monte Carlo calculations of uniquely flexible contributions to virial coefficients

Shaul

Schultz

Kofke

2011

The Journal of Chemical Physics

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We present methods for computing contributions to the virial coefficients uniquely associated with molecular flexibility, and we demonstrate their use with application to the third, fourth, and fifth virial coefficients of united-atom models of linear alkanes and methanol belonging to the suite of transferrable potentials for phase equilibria (TraPPE-UA). We find that these uniquely flexible contributions are more difficult to compute than the remainder of the coefficient, especially for the conditions at which they appear to be most important. The significance of these contributions relative to the full virial coefficient grows with the number of sites (the size of the molecule), the number of molecules, and, to a certain extent, the temperature. The nature of the site-site interactions is of great importance: the significance of the uniquely flexible contribution at third and fourth order is orders of magnitude larger for TraPPE-UA methanol, which has Coulombic interactions, than for TraPPE-UA propane, which does not, even though both models have three sites per molecule and comparable bending potentials. While the uniquely flexible contribution of TraPPE-UA propane has a negligible impact on its third-order virial-equation-of-state estimate of the critical point, the uniquely flexible contribution of TraPPE-UA methanol increases this estimate of its critical pressure by about 5%.

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“…For DHSs in the presence of finite fields, B 3b was found to give a small but significant negative contribution to B 3 on the order of 10% [44]. It is emphasised that these 'fluctuation' corrections to the virial coefficients were first identified by Caracciolo et al as arising from intramolecular degrees of freedom [46,47].…”

Section: Virial Coefficientsmentioning

confidence: 94%

“…A simple resummation of the VE, based on a logarithmic form for the free energy, gives a convenient and accurate expression for the equation of state [44,45]. Along the way, it has been emphasised that the basic definitions of the virial coefficients contain additional terms related to the field; this was anticipated in earlier work by additional terms related to intramolecular degrees of freedom [46,47]. Explicit expressions for the virial coefficients have been developed, and compared with the results from Mayer-sampling calculations [48].…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics of the Stockmayer fluid in an applied field

et al. 2015

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The thermodynamic properties of the Stockmayer fluid in an applied field are studied using theory and computer simulation. Theoretical expressions for the second and third virial coefficients are obtained in terms of the dipolar coupling constant (λ, measuring the strength of dipolar interactions as compared to thermal energy) and dipole-field interaction energy (α, being proportional to the applied field strength). These expressions are tested against numerical results obtained by Mayer sampling calculations. The expression for the second virial coefficient contains terms up to λ 4 , and is found to be accurate over realistic ranges of dipole moment and temperature, and over the entire range of the applied field strength (from zero to infinity). The corresponding expression for the third virial coefficient is truncated at λ 3 , and is not very accurate: higher order terms are very difficult to calculate. The virial coefficients are incorporated in to a thermodynamic theory based on a logarithmic representation of the Helmholtz free energy. This theory is designed to retain the input virial coefficients, and account for some higher order terms in the sense of a resummation. The compressibility factor is obtained from the theory and compared to results from molecular dynamics simulations with a typical value λ = 1. Despite the mathematical approximations of the virial coefficients, the theory captures the effects of the applied field very well. Finally, the vapourliquid critical parameters are determined from the theory, and compared to published simulation results; the agreement between the theory and simulations is good.

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Third Virial Coefficient for 4‐Arm and 6‐Arm Star Polymers

Cited by 15 publications

References 34 publications

Thermodynamics of ferrofluids in applied magnetic fields

Thermodynamics of ferrofluids in applied magnetic fields

Mayer-sampling Monte Carlo calculations of uniquely flexible contributions to virial coefficients

Thermodynamics of the Stockmayer fluid in an applied field

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