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Fused hard sphere chains

2011-03-16T12:27:05Z

193.6.32.101:

[[Image:FHSC_linear.png|Example of the fused hard sphere chain model, shown here in a linear configuration.|thumb|right]]

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)

:<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.

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) ~~~~
\scriptstyle{
L^* \leq 1 ~\and~ \left(\gamma=\pi ~ \or ~
L^* \sin{\frac\gamma{2}} \geq \frac{1}{2}\right)
}
</math>

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>
==Equation of state==
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by

:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>

where

:<math>C_{\rm FHSC} = 5.66\alpha(1-0.045[\alpha-1]^{1/2}\eta_0)</math>

and

:<math>\eta_0(P^*) = \frac{\sqrt{1+4(1+3\alpha)P^*}-1}{2+6\alpha}</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==
*[[Rigid fully flexible fused hard sphere model]]
==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)]
#[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/10.1063/1.1517604 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/10.1063/1.470528 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/10.1063/1.459523 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/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
[[category:liquid crystals]]
[[category:models]]

Fused hard sphere chains

2011-03-16T08:59:06Z

193.6.32.101:

[[Image:FHSC_linear.png|Example of the fused hard sphere chain model, shown here in a linear configuration.|thumb|right]]

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)

:<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.

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>

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>
==Equation of state==
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by

:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>

where

:<math>C_{\rm FHSC} = 5.66\alpha(1-0.045[\alpha-1]^{1/2}\eta_0)</math>

and

:<math>\eta_0(P^*) = \frac{\sqrt{1+4(1+3\alpha)P^*}-1}{2+6\alpha}</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==
*[[Rigid fully flexible fused hard sphere model]]
==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)]
#[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/10.1063/1.1517604 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/10.1063/1.470528 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/10.1063/1.459523 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/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
[[category:liquid crystals]]
[[category:models]]

Hard spherocylinders

2011-03-09T11:12:16Z

193.6.32.101: /* Minimum distance */

[[Image:spherocylinder_purple.png|thumb|right]]
The '''hard spherocylinder''' model consists of an impenetrable cylinder, capped at both ends
by hemispheres whose diameters are the same as the diameter of the cylinder. The hard spherocylinder model
has been studied extensively because of its propensity to form both [[Nematic phase | nematic]] and [[Smectic phases | smectic]] [[Liquid crystals | liquid crystalline]] phases <ref>[http://dx.doi.org/10.1038/332822a0 D. Frenkel, H. N. W. Lekkerkerker and A. Stroobants "Thermodynamic stability of a smectic phase in a system of hard rods", Nature '''332''' p. 822 (1988)]</ref> as well as forming a [[Plastic crystals | plastic crystal]] phase for short <math>L</math> <ref>[http://dx.doi.org/10.1063/1.474626 C. Vega and P. A. Monson "Plastic crystal phases of hard dumbbells and hard spherocylinders", Journal of Chemical Physics '''107''' pp. 2696-2697 (1997)]</ref> . One of the first studies of hard spherocylinders was undertaken by Cotter and Martire <ref>[http://dx.doi.org/10.1063/1.1673232 Martha A. Cotter and Daniel E. Martire "Statistical Mechanics of Rodlike Particles. II. A Scaled Particle Investigation of the Aligned to Isotropic Transition in a Fluid of Rigid Spherocylinders", Journal of Chemical Physics '''52''' pp. 1909-1919 (1970)]</ref> using [[scaled-particle theory]], and one of the first simulations was in the classic work of Jacques Vieillard-Baron <ref>[http://dx.doi.org/10.1080/00268977400102161 Jacques Vieillard-Baron "The equation of state of a system of hard spherocylinders", Molecular Physics '''28''' pp. 809-818 (1974)]</ref>. In the limit of zero diameter the hard spherocylinder becomes a line segment, often known as the [[3-dimensional hard rods |hard rod model]], and in the limit <math>L=0</math> one has the [[hard sphere model]]. A closely related model is that of the [[Oblate hard spherocylinders | oblate hard spherocylinder]].
==Volume==
The molecular volume of the spherocylinder is given by

:<math>v_0 = \pi \left( \frac{LD^2}{4} + \frac{D^3}{6} \right)</math>

where <math>L</math> is the length of the cylindrical part of the spherocylinder and <math>D</math> is the diameter.
==Minimum distance==
The minimum distance between two spherocylinders can be calculated using an algorithm published by Vega and Lago <ref>[http://dx.doi.org/10.1016/0097-8485(94)80023-5 Carlos Vega and Santiago Lago "A fast algorithm to evaluate the shortest distance between rods", Computers & Chemistry '''18''' pp. 55-59 (1994)]</ref>. The [[Source code for the minimum distance between two rods | source code can be found here]]. Such an algorithm is essential in, for example, a [[Monte Carlo]] simulation, in order to check for overlaps between two sites.

The original code did not give the symmetry property of the distance for almost parallel rods. The revised algorithm in C for systems with periodic boundary conditions can be found [[Rev. source code for the minimum distance between two rods in C | here]].

==Virial coefficients==
:''Main article: [[Hard spherocylinders: virial coefficients]]''
==Phase diagram==
:''Main aritcle: [[Phase diagram of the hard spherocylinder model]]''
==See also==
*[[Charged hard spherocylinders]]
*[[Oblate hard spherocylinders]]

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1021/j100303a008 Daan Frenkel "Onsager's spherocylinders revisited", Journal of Physical Chemistry '''91''' pp. 4912-4916 (1987)]
[[Category: Models]]

Rev. source code for the minimum distance between two rods in C

2011-03-09T11:04:16Z

193.6.32.101: Created page with " /* Revision of Carlos Vega & Santiago Lago Computers Chem. 18, 55-59, 1994 Subrutine to evaluate the shortest distance between two rods of different length The o..."

/*
Revision of
Carlos Vega & Santiago Lago
Computers Chem. 18, 55-59, 1994

Subrutine to evaluate the shortest distance between two rods of
different length

The original code did not give the symmetry property of the distance for almost parallel rods.
The coordinates of the centers of the rods should be given in a periodic system

r1,r2: centers of rods
w1,w2: unit orientation vectors of rods
lh1,lh2: halves of the length of rods
Lv.x,Lv.y,Lv.z the edges of the periodic simulation cell
*/

#include <math.h>

//----------------- VECTOR operations: -----------------------------------------------------

#define VECT_COMMA ,
#define VECT_PAR (
#define VECT_PSEQ(_,SEP) (_ x)) SEP (_ y)) SEP (_ z))

#define VECT_COMP(x) .x
#define VECT_OP(A,COMP,OP,x) A COMP(x) OP
#define VECT_A_OP_B(A,OP,B,x) VECT_OP(A,VECT_COMP,OP,x) VECT_OP(B,VECT_COMP,,x)

#define VECT_OSEQ_(A,OP,B,SEP,_) \
VECT_PSEQ(VECT_A_OP_B VECT_PAR A VECT_COMMA OP VECT_COMMA B VECT_COMMA,SEP##_)

#define VECT_OSEQ(A,OP,B,SEP) VECT_OSEQ_(A,OP,B,SEP,)
#define VECT_PROD(A,B) VECT_OSEQ(A,*,B,+) /* product of A and B */
#define VECT_NORM2(A) VECT_PROD(A,A) /* square of the norm of A */

#define VECT_OLIST(A,OP,B) VECT_OSEQ_(A,OP,B,VECT_COMMA,) /* (A.x OP B.x), ... */

#define VECT_SEQ(V,SEP) V(x) SEP V(y) SEP V(z) /* because of the single macro expansion */
#define VECT_LIST(V) VECT_SEQ(V,VECT_COMMA) /* V(x), ... */

typedef struct { double VECT_LIST(); } coo_t;

//---------------------------------------------------------------------------------------

extern coo_t Lv;

// Minimum distance in the periodic system:

#define MIN_RIJ(x) \
( FX= fabs(rij.x),(FX<Lv.x-FX)?rij.x:(rij.x-((rij.x >0)?Lv.x:-Lv.x) ) )

#define PW2(x) (x*x)

static inline double sign(double a,double b) { return a= fabs(a),(b<0)?-a:a; }

//---------------- Distance of two rods: -------------------------------------

double dist2_rods(coo_t r1,coo_t r2,coo_t w1,coo_t w2,double lh1,double lh2)
{
coo_t rij= { VECT_OLIST(r2,-,r1) };
register double FX;
coo_t min_rij= { VECT_LIST(MIN_RIJ) };
double
xla,xmu,
rr= VECT_NORM2(min_rij),
rw1= VECT_PROD(min_rij,w1),
rw2= VECT_PROD(min_rij,w2),
w1w2= VECT_PROD(w1,w2),
cc= 1-PW2(w1w2);

// Checking whether the rods are or not parallel:
// The original code is modified to have symmetry:

if(cc<1e-6) {
if(rw1 && rw2) {
xla= rw1/2;
xmu= -rw2/2;
}
else return rr;
}

else {

// Step 1

xla= (rw1-w1w2*rw2)/cc;
xmu= (-rw2+w1w2*rw1)/cc;
}

// Step 2

if( fabs(xla)>lh1 || fabs(xmu)>lh2 ) {

// Step 3 - 7

if(fabs(xla)-lh1>fabs(xmu)-lh2) {
xla= sign(lh1,xla);
xmu= xla*w1w2-rw2;
if( fabs(xmu)>lh2 ) xmu= sign(lh2,xmu);
}
else {
xmu= sign(lh2,xmu);
xla= xmu*w1w2+rw1;
if( fabs(xla)>lh1 ) xla= sign(lh1,xla);
}
}

// Step 8

return rr+PW2(xla)+PW2(xmu) + 2*(xmu*rw2 -xla*(rw1+xmu*w1w2));
}
==References==
#[http://dx.doi.org/10.1016/0097-8485(94)80023-5 Carlos Vega and Santiago Lago "A fast algorithm to evaluate the shortest distance between rods", Computers & Chemistry '''18''' pp. 55-59 (1994)]

{{Source}}
[[category: C code]]

Fused hard sphere chains

2011-03-08T09:28:22Z

193.6.32.101:

[[Image:FHSC_linear.png|Example of the fused hard sphere chain model, shown here in a linear configuration.|thumb|right]]

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)

:<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.

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>

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>
==Equation of state==
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by

:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>

where

:<math>C_{\rm FHSC} = 5.66\alpha(1-0.045[\alpha-1]^{1/2}\eta_0)</math>

and

:<math>\eta_0(P^*) = \frac{\sqrt{1+4(1+3\alpha)P^*}-1}{2+6\alpha}</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==
*[[Rigid fully flexible fused hard sphere model]]
==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)]
#[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/10.1063/1.1517604 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/10.1063/1.470528 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/10.1063/1.459523 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/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
[[category:liquid crystals]]
[[category:models]]

Fused hard sphere chains

2011-03-08T08:32:47Z

193.6.32.101:

[[Image:FHSC_linear.png|Example of the fused hard sphere chain model, shown here in a linear configuration.|thumb|right]]

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)

:<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.

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>

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>
==Equation of state==
The Vörtler and Nezbeda [[Equations of state | equation of state]] is given by

:<math>Z_{\mathrm{FHSC}}= 1+ (1+3\alpha)\eta_0(P^*) + C_{\rm FHSC}[\eta_0(P^*)]^{1.83}</math>

where

:<math>C_{\rm FHSC} = 5.66\alpha(1-0.045[\alpha-1]^{1/2}\eta_0)</math>

and

:<math>\eta_0(P^*) = \frac{\sqrt{1+4(1+3\alpha)P^*}-1}{2+6\alpha}</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==
*[[Rigid fully flexible fused hard sphere model]]
==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)]
#[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/10.1063/1.1517604 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/10.1063/1.470528 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/10.1063/1.459523 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/10.1080/002689798168989 Antoine Chamoux and Aurelien Perera "On the linear hard sphere chain fluids", Molecular Physics '''93'' pp. 649-661 (1998)]
[[category:liquid crystals]]
[[category:models]]