Fused hard sphere chains: Difference between revisions

Revision as of 13:27, 16 March 2011

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)~~~~\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. 5 Eq. 12)

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

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}

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 }}

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)
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)

@@ Line 11: / Line 11: @@
 The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)
-:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6}  \left( 1 + (m-1)\frac{3L^*  - L^{*3}}{2} \right)  ~~~~ L^* \leq 1 </math>
+:<math>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)
+}
+</math>
-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. 5 Eq. 12)
 :<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>