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Equations of state for hard spheres

2018-12-18T13:20:14Z

Smwaziri:

The following is a list of [[equations of state]] designed for the [[hard sphere model]]:
*[[Carnahan-Starling equation of state]]
*[[Equations of state for crystals of hard spheres]]
*[[Hard hypersphere equation of state |Hard hyperspheres]]
*[[Kolafa-Labík-Malijevský equation of state]]
*[[Santos-Lopez de Haro hard sphere equation of state]]
*[[Hansen-Goos hard sphere equation of state]]
*[[Equations of state for crystals of hard spheres]]
*[[WC1 and WC2 hard sphere equations of state]]
*[[Hamad hard sphere equation of state]]
==See also==
*[[Exact solution of the Percus Yevick integral equation for hard spheres]]
*[[Equations of state for hard sphere mixtures]]
*[[Equations of state for hard disks]]
==Related reading==
*[http://dx.doi.org/10.1007/978-3-540-78767-9_3 A. Mulero, C.A. Galán, M.I. Parra and F. Cuadros "Equations of State for Hard Spheres and Hard Disks", Lecture Notes in Physics '''753''' Chapter 3 pp.37-109 (2008)]
[[category: equations of state]]

Fused hard sphere chains

2018-12-18T09:35:14Z

Smwaziri: /* Equation of state */

[[Image:FHSC_linear.png|Example of the fused hard sphere chain model, shown here in a linear configuration.|thumb|right]]

In the '''fused hard sphere chain''' model the ''molecule'' is built up form a string of overlapping [[hard sphere model|hard sphere sites]], each of diameter <math>\sigma</math>.

An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. 4 Eq. 2.18)

:<math>m_{\rm effective} = \frac{[1+(m-1)L^*]^3}{[1+(m-1)L^*(3-L^{*2})/2]^2}</math>

where ''m'' is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.

The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)

:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6} \left( 1 + (m-1)\frac{3L^* - L^{*3}}{2} \right) ~~~~
\scriptstyle{
L^* \leq 1 ~\and~ \left(\gamma=\pi ~ \or ~
L^* \sin{\frac\gamma{2}} \geq \frac{1}{2}\right)
}
</math>

where <math>0<\gamma \leq \pi</math> is the minimal bond angle, and the surface area is given by (Ref. 5 Eq. 12)

:<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>
==Equation of state==
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by

:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>

where

:<math>C_{\rm FHSC} = 5.66\alpha(1-0.045[\alpha-1]^{1/2}\eta_0)</math>

and

:<math>\eta_0(P^*) = \frac{\sqrt{1+4(1+3\alpha)P^*}-1}{2+6\alpha}</math>

The Waziri and Hamad [[Equations of state | equation of state]] for fused hard sphere chain fluids is given by

:<math>Z_{\mathrm{FHSC}} = 1 + 4m_{\mathrm{eff}}P^{*} + \frac{3}{4}m_{\mathrm{eff}}P^{*}\ln\left[\frac{3+P^{*}}{3+25P^{*}}\right] + \frac{216(m_{\mathrm{eff}} - 1)P^{*}}{(3+P^{*})(3+25P^{*})\{16+3\ln[(3+P^{*})/(3+25P^{*})]\}}</math>

where

:<math>m_{\mathrm{eff}}=\frac{2+3(m-1)L^{*}+2(m-1)^{2}L^{*2}+(m-1)L^{*3}}{2+3(m-1)L^{*}-(m-1)L^{*3}}</math>

#Horst L. Vörtler and I. Nezbeda "Volume-explicit equation of state and excess volume of mixing of fused hard sphere fluids", Berichte der Bunsen-Gesellschaft '''94''' pp. 559- (1990)
#[http://dx.doi.org/10.1021/ie800755s Saidu M. Waziri and Esam Z. Hamad "Volume-Explicit Equation of State for Fused Hard Sphere Chain Fluids", Industrial & Engineering Chemistry Research '''47''' pp. 9658-9662 (2008)]

==See also==
*[[Rigid fully flexible fused hard sphere model]]
==References==
#[http://dx.doi.org/10.1080/00268979100100191 M. Whittle and A. J. Masters "Liquid crystal formation in a system of fused hard spheres", Molecular Physics '''72''' pp. 247-265 (1991)]
#[http://dx.doi.org/10.1103/PhysRevE.64.011703 Carl McBride, Carlos Vega, and Luis G. MacDowell "Isotropic-nematic phase transition: Influence of intramolecular flexibility using a fused hard sphere model" Physical Review E '''64''' 011703 (2001)]
#[http://dx.doi.org/10.1063/1.1517604 Carl McBride and Carlos Vega "A Monte Carlo study of the influence of molecular flexibility on the phase diagram of a fused hard sphere model", Journal of Chemical Physics '''117''' pp. 10370-10379 (2002)]
#[http://dx.doi.org/10.1063/1.470528 Yaoqi Zhou, Carol K. Hall and George Stell "Thermodynamic perturbation theory for fused hard-sphere and hard-disk chain fluids", Journal of Chemical Physics '''103''' pp. 2688-2695 (1995)]
#[http://dx.doi.org/10.1063/1.459523 T. Boublík, C. Vega, and M. Diaz-Peña "Equation of state of chain molecules", Journal of Chemical Physics '''93''' pp. pp. 730-736 (1990)]
#[http://dx.doi.org/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
[[category:liquid crystals]]
[[category:models]]

Fused hard sphere chains

2018-12-18T09:33:59Z

Smwaziri: /* Equation of state */

[[Image:FHSC_linear.png|Example of the fused hard sphere chain model, shown here in a linear configuration.|thumb|right]]

In the '''fused hard sphere chain''' model the ''molecule'' is built up form a string of overlapping [[hard sphere model|hard sphere sites]], each of diameter <math>\sigma</math>.

An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. 4 Eq. 2.18)

:<math>m_{\rm effective} = \frac{[1+(m-1)L^*]^3}{[1+(m-1)L^*(3-L^{*2})/2]^2}</math>

where ''m'' is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.

The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)

:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6} \left( 1 + (m-1)\frac{3L^* - L^{*3}}{2} \right) ~~~~
\scriptstyle{
L^* \leq 1 ~\and~ \left(\gamma=\pi ~ \or ~
L^* \sin{\frac\gamma{2}} \geq \frac{1}{2}\right)
}
</math>

where <math>0<\gamma \leq \pi</math> is the minimal bond angle, and the surface area is given by (Ref. 5 Eq. 12)

:<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>
==Equation of state==
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by

:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>

where

:<math>C_{\rm FHSC} = 5.66\alpha(1-0.045[\alpha-1]^{1/2}\eta_0)</math>

and

:<math>\eta_0(P^*) = \frac{\sqrt{1+4(1+3\alpha)P^*}-1}{2+6\alpha}</math>

The Waziri and Hamad [[Equations of state | equation of state]] for fused hard sphere chain fluids is given by

:<math>Z_{FHSC} = 1 + 4m_{\mathrm{eff}}P^{*} + \frac{3}{4}m_{\mathrm{eff}}P^{*}\ln\left[\frac{3+P^{*}}{3+25P^{*}}\right] + \frac{216(m_{\mathrm{eff}} - 1)P^{*}}{(3+P^{*})(3+25P^{*})\{16+3\ln[(3+P^{*})/(3+25P^{*})]\}}</math>

where

:<math>m_{\mathrm{eff}}=\frac{2+3(m-1)L^{*}+2(m-1)^{2}L^{*2}+(m-1)L^{*3}}{2+3(m-1)L^{*}-(m-1)L^{*3}}</math>

#Horst L. Vörtler and I. Nezbeda "Volume-explicit equation of state and excess volume of mixing of fused hard sphere fluids", Berichte der Bunsen-Gesellschaft '''94''' pp. 559- (1990)
#[http://dx.doi.org/10.1021/ie800755s Saidu M. Waziri and Esam Z. Hamad "Volume-Explicit Equation of State for Fused Hard Sphere Chain Fluids", Industrial & Engineering Chemistry Research '''47''' pp. 9658-9662 (2008)]

==See also==
*[[Rigid fully flexible fused hard sphere model]]
==References==
#[http://dx.doi.org/10.1080/00268979100100191 M. Whittle and A. J. Masters "Liquid crystal formation in a system of fused hard spheres", Molecular Physics '''72''' pp. 247-265 (1991)]
#[http://dx.doi.org/10.1103/PhysRevE.64.011703 Carl McBride, Carlos Vega, and Luis G. MacDowell "Isotropic-nematic phase transition: Influence of intramolecular flexibility using a fused hard sphere model" Physical Review E '''64''' 011703 (2001)]
#[http://dx.doi.org/10.1063/1.1517604 Carl McBride and Carlos Vega "A Monte Carlo study of the influence of molecular flexibility on the phase diagram of a fused hard sphere model", Journal of Chemical Physics '''117''' pp. 10370-10379 (2002)]
#[http://dx.doi.org/10.1063/1.470528 Yaoqi Zhou, Carol K. Hall and George Stell "Thermodynamic perturbation theory for fused hard-sphere and hard-disk chain fluids", Journal of Chemical Physics '''103''' pp. 2688-2695 (1995)]
#[http://dx.doi.org/10.1063/1.459523 T. Boublík, C. Vega, and M. Diaz-Peña "Equation of state of chain molecules", Journal of Chemical Physics '''93''' pp. pp. 730-736 (1990)]
#[http://dx.doi.org/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
[[category:liquid crystals]]
[[category:models]]

Fused hard sphere chains

2018-12-17T14:10:01Z

Smwaziri: /* Equation of state */

[[Image:FHSC_linear.png|Example of the fused hard sphere chain model, shown here in a linear configuration.|thumb|right]]

In the '''fused hard sphere chain''' model the ''molecule'' is built up form a string of overlapping [[hard sphere model|hard sphere sites]], each of diameter <math>\sigma</math>.

An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. 4 Eq. 2.18)

:<math>m_{\rm effective} = \frac{[1+(m-1)L^*]^3}{[1+(m-1)L^*(3-L^{*2})/2]^2}</math>

where ''m'' is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.

The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)

:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6} \left( 1 + (m-1)\frac{3L^* - L^{*3}}{2} \right) ~~~~
\scriptstyle{
L^* \leq 1 ~\and~ \left(\gamma=\pi ~ \or ~
L^* \sin{\frac\gamma{2}} \geq \frac{1}{2}\right)
}
</math>

where <math>0<\gamma \leq \pi</math> is the minimal bond angle, and the surface area is given by (Ref. 5 Eq. 12)

:<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>
==Equation of state==
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by

:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>

where

:<math>C_{\rm FHSC} = 5.66\alpha(1-0.045[\alpha-1]^{1/2}\eta_0)</math>

and

:<math>\eta_0(P^*) = \frac{\sqrt{1+4(1+3\alpha)P^*}-1}{2+6\alpha}</math>

The Waziri and Hamad [[Equations of state | equation of state]] is given by

:<math>Z_{FHS} = 1 + 4m_{\mathrm{eff}}P^{*} + \frac{3}{4}m_{\mathrm{eff}}P^{*}\ln\left[\frac{3+P^{*}}{3+25P^{*}}\right] + \frac{216(m_{\mathrm{eff}} - 1)P^{*}}{(3+P^{*})(3+25P^{*})\{16+3\ln[(3+P^{*})/(3+25P^{*})]\}}</math>

where

:<math>m_{\mathrm{eff}}=\frac{2+3(m-1)L^{*}+2(m-1)^{2}L^{*2}+(m-1)L^{*3}}{2+3(m-1)L^{*}-(m-1)L^{*3}}</math>

#Horst L. Vörtler and I. Nezbeda "Volume-explicit equation of state and excess volume of mixing of fused hard sphere fluids", Berichte der Bunsen-Gesellschaft '''94''' pp. 559- (1990)
#[http://dx.doi.org/10.1021/ie800755s Saidu M. Waziri and Esam Z. Hamad "Volume-Explicit Equation of State for Fused Hard Sphere Chain Fluids", Industrial & Engineering Chemistry Research '''47''' pp. 9658-9662 (2008)]

==See also==
*[[Rigid fully flexible fused hard sphere model]]
==References==
#[http://dx.doi.org/10.1080/00268979100100191 M. Whittle and A. J. Masters "Liquid crystal formation in a system of fused hard spheres", Molecular Physics '''72''' pp. 247-265 (1991)]
#[http://dx.doi.org/10.1103/PhysRevE.64.011703 Carl McBride, Carlos Vega, and Luis G. MacDowell "Isotropic-nematic phase transition: Influence of intramolecular flexibility using a fused hard sphere model" Physical Review E '''64''' 011703 (2001)]
#[http://dx.doi.org/10.1063/1.1517604 Carl McBride and Carlos Vega "A Monte Carlo study of the influence of molecular flexibility on the phase diagram of a fused hard sphere model", Journal of Chemical Physics '''117''' pp. 10370-10379 (2002)]
#[http://dx.doi.org/10.1063/1.470528 Yaoqi Zhou, Carol K. Hall and George Stell "Thermodynamic perturbation theory for fused hard-sphere and hard-disk chain fluids", Journal of Chemical Physics '''103''' pp. 2688-2695 (1995)]
#[http://dx.doi.org/10.1063/1.459523 T. Boublík, C. Vega, and M. Diaz-Peña "Equation of state of chain molecules", Journal of Chemical Physics '''93''' pp. pp. 730-736 (1990)]
#[http://dx.doi.org/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
[[category:liquid crystals]]
[[category:models]]

Fused hard sphere chains

2018-12-17T14:05:50Z

Smwaziri: /* Equation of state */

[[Image:FHSC_linear.png|Example of the fused hard sphere chain model, shown here in a linear configuration.|thumb|right]]

In the '''fused hard sphere chain''' model the ''molecule'' is built up form a string of overlapping [[hard sphere model|hard sphere sites]], each of diameter <math>\sigma</math>.

An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. 4 Eq. 2.18)

:<math>m_{\rm effective} = \frac{[1+(m-1)L^*]^3}{[1+(m-1)L^*(3-L^{*2})/2]^2}</math>

where ''m'' is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.

The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)

:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6} \left( 1 + (m-1)\frac{3L^* - L^{*3}}{2} \right) ~~~~
\scriptstyle{
L^* \leq 1 ~\and~ \left(\gamma=\pi ~ \or ~
L^* \sin{\frac\gamma{2}} \geq \frac{1}{2}\right)
}
</math>

where <math>0<\gamma \leq \pi</math> is the minimal bond angle, and the surface area is given by (Ref. 5 Eq. 12)

:<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>
==Equation of state==
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by

:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>

where

:<math>C_{\rm FHSC} = 5.66\alpha(1-0.045[\alpha-1]^{1/2}\eta_0)</math>

and

:<math>\eta_0(P^*) = \frac{\sqrt{1+4(1+3\alpha)P^*}-1}{2+6\alpha}</math>

The Waziri and Hamad [[Equations of state | equation of state]] is given by

:<math>Z_{FHS} = 1 + 4m_{\mathrm{eff}}P^{*} + \frac{3}{4}m_{\mathrm{eff}}P^{*}\ln\left[\frac{3+P^{*}}{3+25P^{*}}\right] + \frac{216(m_{\mathrm{eff}} - 1)P^{*}}{(3+P^{*})(3+25P^{*})\{16+3\ln[(3+P^{*})/(3+25P^{*})]\}}</math>

where

:<math>m_{\mathrm{eff}}=\frac{2+3(m-1)l^{*}+2(m-1)^{2}l^{*2}+(m-1)l^{*3}}{2+3(m-1)l^{*}-(m-1)l^{*3}}</math>

#Horst L. Vörtler and I. Nezbeda "Volume-explicit equation of state and excess volume of mixing of fused hard sphere fluids", Berichte der Bunsen-Gesellschaft '''94''' pp. 559- (1990)
#[http://dx.doi.org/10.1021/ie800755s Saidu M. Waziri and Esam Z. Hamad "Volume-Explicit Equation of State for Fused Hard Sphere Chain Fluids", Industrial & Engineering Chemistry Research '''47''' pp. 9658-9662 (2008)]

==See also==
*[[Rigid fully flexible fused hard sphere model]]
==References==
#[http://dx.doi.org/10.1080/00268979100100191 M. Whittle and A. J. Masters "Liquid crystal formation in a system of fused hard spheres", Molecular Physics '''72''' pp. 247-265 (1991)]
#[http://dx.doi.org/10.1103/PhysRevE.64.011703 Carl McBride, Carlos Vega, and Luis G. MacDowell "Isotropic-nematic phase transition: Influence of intramolecular flexibility using a fused hard sphere model" Physical Review E '''64''' 011703 (2001)]
#[http://dx.doi.org/10.1063/1.1517604 Carl McBride and Carlos Vega "A Monte Carlo study of the influence of molecular flexibility on the phase diagram of a fused hard sphere model", Journal of Chemical Physics '''117''' pp. 10370-10379 (2002)]
#[http://dx.doi.org/10.1063/1.470528 Yaoqi Zhou, Carol K. Hall and George Stell "Thermodynamic perturbation theory for fused hard-sphere and hard-disk chain fluids", Journal of Chemical Physics '''103''' pp. 2688-2695 (1995)]
#[http://dx.doi.org/10.1063/1.459523 T. Boublík, C. Vega, and M. Diaz-Peña "Equation of state of chain molecules", Journal of Chemical Physics '''93''' pp. pp. 730-736 (1990)]
#[http://dx.doi.org/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
[[category:liquid crystals]]
[[category:models]]