Fused hard sphere chains - Revision history

Carl McBride: Cleaned up the references section

2018-12-23T18:45:25Z

Cleaned up the references section

← Older revision		Revision as of 20:45, 23 December 2018
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	In the '''fused hard sphere chain''' model the ''molecule'' is built up form a string of overlapping [[hard sphere model\|hard sphere sites]], each of diameter <math>\sigma</math>.		In the '''fused hard sphere chain''' model the ''molecule'' is built up form a string of overlapping [[hard sphere model\|hard sphere sites]], each of diameter <math>\sigma</math>.

	An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. 4 Eq. 2.18)		An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. <ref>[http://dx.doi.org/10.1063/1.470528 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)]</ref> Eq. 2.18)

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

	where ''m'' is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.		where <math>m</math> is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.

	The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)		The volume of the fused hard sphere chain is given by (Ref. <ref name="BVD">[http://dx.doi.org/10.1063/1.459523 T. Boublík, C. Vega, and M. Diaz-Peña "Equation of state of chain molecules", Journal of Chemical Physics '''93''' pp. pp. 730-736 (1990)]</ref> Eq. 13)

	:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6} \left( 1 + (m-1)\frac{3L^* - L^{*3}}{2} \right) ~~~~		:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6} \left( 1 + (m-1)\frac{3L^* - L^{*3}}{2} \right) ~~~~
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	</math>		</math>

	where <math>0<\gamma \leq \pi</math> is the minimal bond angle, 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.<ref name="BVD" /> Eq. 12)

	:<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>		:<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>
	==Equation of state==		==Equation of state==
	The Vörtler and Nezbeda [[Equations of state \| equation of state]] is given by		The Vörtler and Nezbeda [[Equations of state \| equation of state]] is given by <ref>[https://doi.org/10.1002/bbpc.19900940505 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. 559-563 (1990)]</ref>

	:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^) + C_{\rm FHSC}[\eta_0(P^)]^{1.83}</math>		:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^) + C_{\rm FHSC}[\eta_0(P^)]^{1.83}</math>
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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]] for fused hard sphere chain fluids is given by		The Waziri and Hamad [[Equations of state \| equation of state]] for fused hard sphere chain fluids is given by <ref>[http://dx.doi.org/10.1021/ie800755s 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)]</ref>

	:<math>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^{*})]\}}</math>		:<math>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^{*})]\}}</math>
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	:<math>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}}</math>		:<math>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}}</math>


	~~#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)~~
	#[http://dx.doi.org/10.1021/ie800755s 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==		==See also==
	*[[Rigid fully flexible fused hard sphere model]]		*[[Rigid fully flexible fused hard sphere model]]
	==References==		==References==
	#[http://dx.doi.org/10.1080/00268979100100191 M. Whittle and A. J. Masters "Liquid crystal formation in a system of fused hard spheres", Molecular Physics '''72''' pp. 247-265 (1991)]		<references/>
	#[http://dx.doi.org/10.1103/PhysRevE.64.011703 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)]		;Related reading
	#[http://dx.doi.org/10.1063/1.1517604 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)]		*[http://dx.doi.org/10.1080/00268979100100191 M. Whittle and A. J. Masters "Liquid crystal formation in a system of fused hard spheres", Molecular Physics '''72''' pp. 247-265 (1991)]
	#[http://dx.doi.org/10.1063/1.470528 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)]		*[http://dx.doi.org/10.1103/PhysRevE.64.011703 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)]
	~~#[http://dx.doi.org/10.1063/1.459523 T. Boublík, C. Vega, and M. Diaz-Peña "Equation of state of chain molecules", Journal of Chemical Physics '''93''' pp. pp. 730-736 (1990)]~~		*[http://dx.doi.org/10.1063/1.1517604 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)]
	#[http://dx.doi.org/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]		*[http://dx.doi.org/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
	[[category:liquid crystals]]		[[category:liquid crystals]]
	[[category:models]]		[[category:models]]

Smwaziri: /* Equation of state */

2018-12-18T09:35:14Z

Equation of state

← Older revision		Revision as of 11:35, 18 December 2018
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	The Waziri and Hamad [[Equations of state \| equation of state]] for fused hard sphere chain fluids is given by		The Waziri and Hamad [[Equations of state \| equation of state]] for fused hard sphere chain fluids is given by

	:<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>		:<math>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^{*})]\}}</math>

	where		where

Smwaziri: /* Equation of state */

2018-12-18T09:33:59Z

Equation of state

← Older revision		Revision as of 11:33, 18 December 2018
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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>

	where		where

Smwaziri: /* Equation of state */

2018-12-17T14:10:01Z

Equation of state

← Older revision		Revision as of 16:10, 17 December 2018
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	where		where

	:<math>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}}</math>		:<math>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}}</math>

Smwaziri: /* Equation of state */

2018-12-17T14:05:50Z

Equation of state

← Older revision		Revision as of 16:05, 17 December 2018
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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

			:<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>

			where

			:<math>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}}</math>


	#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)		#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)

193.6.32.101 at 12:27, 16 March 2011

2011-03-16T12:27:05Z

← Older revision		Revision as of 14:27, 16 March 2011
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	The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)		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>		:<math>S_{\mathrm FHSC} = \pi \sigma^2 \left( 1+\left( m-1 \right) L^* \right)</math>

193.6.32.101 at 08:59, 16 March 2011

2011-03-16T08:59:06Z

← Older revision		Revision as of 10:59, 16 March 2011
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	The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)		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( ~~\frac{~~1~~}{2}~~ (m-1)(3L^* - L^{3}~~) +1~~ \right) ~~~~ L^ ~~> 0.5~~ </math>		:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6} \left( 1 + (m-1)\frac{3L^* - L^{3}}{2} \right) ~~~~ L^ \leq 1 </math>

	and the surface area is given by (Ref. 5 Eq. 12)		and the surface area is given by (Ref. 5 Eq. 12)

Carl McBride: Added previously missing condition for the volume.

2011-03-08T10:08:40Z

Added previously missing condition for the volume.

← Older revision		Revision as of 12:08, 8 March 2011
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	The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)		The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)

	:<math>V_{\rm FHSC} =\frac{~~1}{6}~~ \pi \sigma^3 \left( 1 + (m-1)~~\frac{L~~^* ~~\left(3~~-L^{*2}~~\right~~)~~}{2}~~ \right)</math>		:<math>V_{\rm FHSC} =\frac{\pi \sigma^3}{6} \left( \frac{1}{2} (m-1)(3L^* - L^{3}) +1 \right) ~~~~ L^ > 0.5 </math>

	and the surface area is given by (Ref. 5 Eq. 12)		and the surface area is given by (Ref. 5 Eq. 12)

193.6.32.101 at 09:28, 8 March 2011

2011-03-08T09:28:22Z

← Older revision		Revision as of 11:28, 8 March 2011
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	An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. 4 Eq. 2.18)		An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. 4 Eq. 2.18)

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

	where ''m'' is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.		where ''m'' is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.
Line 11:		Line 11:
	The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)		The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)

	:<math>V_{\rm FHSC} =\frac{1}{6} \pi \sigma^3 \left( 1 + (m-1)L^* \left(3-L^{*2}/2\right) \right)</math>		:<math>V_{\rm FHSC} =\frac{1}{6} \pi \sigma^3 \left( 1 + (m-1)\frac{L^* \left(3-L^{*2}\right)}{2} \right)</math>

	and the surface area is given by (Ref. 5 Eq. 12)		and the surface area is given by (Ref. 5 Eq. 12)

193.6.32.101 at 08:32, 8 March 2011

2011-03-08T08:32:47Z

← Older revision		Revision as of 10:32, 8 March 2011
Line 5:		Line 5:
	An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. 4 Eq. 2.18)		An effective number of monomers can be applied to the fused hard sphere chain model by using the relarion (Ref. 4 Eq. 2.18)

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

	where ''m'' is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.		where ''m'' is the number of monomer units in the model, and <math>L^*=L/\sigma</math> is the reduced bond length.
Line 11:		Line 11:
	The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)		The volume of the fused hard sphere chain is given by (Ref. 5 Eq. 13)

	:<math>V_{\rm FHSC} =\frac{1}{6} \pi \sigma^3 \left( 1 + ~~\frac{~~m-1~~}{2}~~L^* \left(3-L^{*2}\right) \right)</math>		:<math>V_{\rm FHSC} =\frac{1}{6} \pi \sigma^3 \left( 1 + (m-1)L^* \left(3-L^{*2}/2\right) \right)</math>

	and the surface area is given by (Ref. 5 Eq. 12)		and the surface area is given by (Ref. 5 Eq. 12)