Difference between revisions of "Fused hard sphere chains"

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m (Equation of state)
m (Equation of state)
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:<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]] is given by
The Waziri and Hamad [[Equations of state | equation of state]] for fused hard sphere chain fluids 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>
:<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>

Revision as of 10:33, 18 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. 4 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. 5 Eq. 13)

V_{\rm FHSC} =\frac{\pi \sigma^3}{6}  \left( 1 + (m-1)\frac{3L^*  - L^{*3}}{2} \right)  ~~~~ 
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. 5 Eq. 12)

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

Equation of state

The Vörtler and Nezbeda equation of state is given by

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


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


\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

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^{*})]\}}



  1. 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)
  2. 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


  1. M. Whittle and A. J. Masters "Liquid crystal formation in a system of fused hard spheres", Molecular Physics 72 pp. 247-265 (1991)
  2. 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)
  3. 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)
  4. 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)
  5. T. Boublík, C. Vega, and M. Diaz-Peña "Equation of state of chain molecules", Journal of Chemical Physics 93 pp. pp. 730-736 (1990)
  6. Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '93 pp. 649-661 (1998)