Difference between revisions of "Fused hard sphere chains"

From SklogWiki
Jump to: navigation, search
m (Equation of state)
(Cleaned up the references section)
 
Line 3: Line 3:
 
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>.
 
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)
+
An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. <ref>[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)]</ref> Eq. 2.18)
  
 
:<math>m_{\rm effective} = \frac{[1+(m-1)L^*]^3}{[1+(m-1)L^*(3-L^{*2})/2]^2}</math>
 
:<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.  
+
where <math>m</math> 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)
+
The volume of the fused hard sphere chain is given by (Ref. <ref name="BVD">[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)]</ref> Eq. 13)
  
 
:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6}  \left( 1 + (m-1)\frac{3L^*  - L^{*3}}{2} \right)  ~~~~  
 
:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6}  \left( 1 + (m-1)\frac{3L^*  - L^{*3}}{2} \right)  ~~~~  
Line 18: Line 18:
 
</math>
 
</math>
  
where <math>0<\gamma \leq \pi</math> is the minimal bond angle, and the surface area is given by (Ref. 5 Eq. 12)
+
where <math>0<\gamma \leq \pi</math> is the minimal bond angle, and the surface area is given by (Ref.<ref name="BVD" />  Eq. 12)
  
 
:<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>
 
:<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>
 
==Equation of state==
 
==Equation of state==
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by
+
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by <ref>[https://doi.org/10.1002/bbpc.19900940505 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. 559-563 (1990)]</ref>
  
 
:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>
 
:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>
Line 34: Line 34:
 
:<math>\eta_0(P^*) = \frac{\sqrt{1+4(1+3\alpha)P^*}-1}{2+6\alpha}</math>
 
:<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
+
The Waziri and Hamad [[Equations of state | equation of state]] for fused hard sphere chain fluids is given by <ref>[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)]</ref>
  
 
:<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>
 
:<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>
Line 41: Line 41:
  
 
:<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>
 
:<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==
 
==See also==
 
*[[Rigid fully flexible fused hard sphere model]]
 
*[[Rigid fully flexible fused hard sphere model]]
 
==References==
 
==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)]
+
<references/>
#[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)]
+
;Related reading
#[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/10.1080/00268979100100191 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/10.1063/1.470528    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/10.1103/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/10.1063/1.459523    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/10.1063/1.1517604      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/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
+
*[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:liquid crystals]]
 
[[category:models]]
 
[[category:models]]

Latest revision as of 20:45, 23 December 2018

Example of the fused hard sphere chain model, shown here in a linear configuration.

In the fused hard sphere chain model the molecule is built up form a string of overlapping hard sphere sites, each of diameter \sigma.

An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. [1] Eq. 2.18)

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

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

The volume of the fused hard sphere chain is given by (Ref. [2] Eq. 13)

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)
}

where 0<\gamma \leq \pi is the minimal bond angle, and the surface area is given by (Ref.[2] Eq. 12)

S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)

Equation of state[edit]

The Vörtler and Nezbeda equation of state is given by [3]

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

where

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

and

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

The Waziri and Hamad equation of state for fused hard sphere chain fluids is given by [4]

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^{*})]\}}

where

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}}

See also[edit]

References[edit]

Related reading