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)

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

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. Eq. 13)

$V_{\rm {FHSC}}={\frac {\pi \sigma ^{3}}{6}}\left(1+(m-1){\frac {3L^{*}-L^{*3}}{2}}\right)~~~~\scriptstyle {L^{*}\leq 1~\land ~\left(\gamma =\pi ~\lor ~L^{*}\sin {\frac {\gamma }{2}}\geq {\frac {1}{2}}\right)}$

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{\rm FHSC} = 5.66\alpha(1-0.045[\alpha-1]^{1/2}\eta_0)}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

• 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)