# Fused hard sphere chains

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

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

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

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

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

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

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

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

## Equation of state

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

${\displaystyle Z_{\mathrm {FHSC} }=1+(1+3\alpha )\eta _{0}(P^{*})+C_{\rm {FHSC}}[\eta _{0}(P^{*})]^{1.83}}$

where

${\displaystyle C_{\rm {FHSC}}=5.66\alpha (1-0.045[\alpha -1]^{1/2}\eta _{0})}$

and

${\displaystyle \eta _{0}(P^{*})={\frac {{\sqrt {1+4(1+3\alpha )P^{*}}}-1}{2+6\alpha }}}$

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

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

where

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