Fused hard sphere chains

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. Eq. 2.18)

$m_{\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}=\frac{[1+(m-1)L^{*}{]}^3}{[1+(m-1)L^{*}(3-L^{*2})/2{]}^2}$

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. Eq. 13)

$V_{\mathrm{F}\mathrm{H}\mathrm{S}\mathrm{C}}=\frac{\pi {\sigma }^3}{6}(1+(m-1)\frac{3L^{*}-L^{*3}}{2})L^{*}\le 1\wedge (\gamma =\pi \vee L^{*}\sin \frac{\gamma }{2}\ge \frac{1}{2})$

where $0<\gamma \le \pi$ is the minimal bond angle, and the surface area is given by (Ref. Eq. 12)

$S_{\mathrm{F}HSC}=\pi {\sigma }^2(1+(m-1)L^{*})$

Equation of state

The Vörtler and Nezbeda equation of state is given by

$Z_{\mathrm{F}\mathrm{H}\mathrm{S}\mathrm{C}}=1+(1+3\alpha ){\eta }_0(P^{*})+C_{\mathrm{F}\mathrm{H}\mathrm{S}\mathrm{C}}[{\eta }_0(P^{*}){]}^{1.83}$

where

$C_{\mathrm{F}\mathrm{H}\mathrm{S}\mathrm{C}}=5.66\alpha (1-0.045[\alpha -1{]}^{1/2}{\eta }_0)$

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

$Z_{\mathrm{F}\mathrm{H}\mathrm{S}\mathrm{C}}=1+4m_{\mathrm{e}\mathrm{f}\mathrm{f}}{P}^{*}+\frac{3}{4}{m}_{\mathrm{e}\mathrm{f}\mathrm{f}}{P}^{*}\ln [\frac{3+P^{*}}{3+25P^{*}}]+\frac{216(m_{\mathrm{e}\mathrm{f}\mathrm{f}}-1)P^{*}}{(3+P^{*})(3+25P^{*})\{16+3\ln [(3+P^{*})/(3+25P^{*})]\}}$

where

$m_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{2+3(m-1)L^{*}+2(m-1{)}^2{L}^{*2}+(m-1)L^{*3}}{2+3(m-1)L^{*}-(m-1)L^{*3}}$

See also

• Rigid fully flexible fused hard sphere model

References

Related reading

• M. Whittle and A. J. Masters "Liquid crystal formation in a system of fused hard spheres", Molecular Physics 72 pp. 247-265 (1991)
• 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)
• 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)
• Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '93 pp. 649-661 (1998)