C60: Difference between revisions

Latest revision as of 18:01, 3 January 2018

C₆₀, also known as Buckminsterfullerene is composed of carbon atoms.

<jmol>

 <jmolApplet>
 <script>set spin X 10; spin on</script>
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   <wikiPageContents>C60.pdb</wikiPageContents>
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</jmol>

C₆₀

Models[edit]

Girifalco potential[edit]

The Girifalco intermolecular pair potential is given by ^[1] (Eq. 4):

\Phi (r)=-\alpha \left[{\frac {1}{s(s-1)^{3}}}+{\frac {1}{s(s+1)^{3}}}-{\frac {2}{s^{4}}}\right]+\beta \left[{\frac {1}{s(s-1)^{9}}}+{\frac {1}{s(s+1)^{9}}}-{\frac {2}{s^{10}}}\right]

where

s={\frac {r}{2a}}

\alpha ={\frac {N^{2}A}{12(2a)^{6}}}

\beta ={\frac {N^{2}B}{90(2a)^{12}}}

where $N$ is the number of atoms on each sphere, i.e. N=60.

Approximate non-conformal potential[edit]

The Approximate non-conformal potential (ANC) for the C60 fullerene is given by (Eq 6 in ^[2]):

\Phi _{12}(z)=\epsilon \left[{\frac {1-a}{(z^{3}/S+1-1/S)^{1/3}-a}}\right]^{12}-2\epsilon \left[{\frac {1-a}{(z^{3}/S+1-1/S)^{1/3}-a}}\right]^{6}

where

$z:=r/r_{m}$
$r_{m}$ = 1.0281 nm
$a$ = 0.09574 is the hard-core diameter in units of $r_{m}$
$\epsilon$ = 3297.28 K is the well depth
$S$ = 0.4120 is a softness parameter

Phase diagram[edit]

^[3] ^[4] ^[5] ^[6]

Liquid phase[edit]

^[7]

Gel phase[edit]

Simulations of the Girifalco potential indicate a possible gel composed solely of C₆₀ molecules ^[8]

References[edit]

Related reading

[1] L. A. Girifalco "Molecular properties of fullerene in the gas and solid phases", Journal of Physical Chemistry 96 pp. 858-861 (1992)

[2] Jesús Eloy Ramos "Effective intermolecular potential and critical point for C60 molecule", Chemical Physics 492 pp. 5-11 (2017)

[3] Ailan Cheng, Michael L. Klein and Carlo Caccamo "Prediction of the phase diagram of rigid C60 molecules", Physical Review Letters 71 pp. 1200-1203 (1993)

[4] L. Mederos and G. Navascués "High-temperature phase diagram of the fullerene C60" Physical Review B 50 pp. 1301-1304 (1994)

[5] M. Hasegawa and K. Ohno "Monte Carlo simulation study of the high-temperature phase diagram of model C60 molecules", Journal of Chemical Physics 111 pp. 5955- (1999)

[6] Pedro Orea "Phase diagrams of model C60 and C70 fullerenes from short-range attractive potentials", Journal of Chemical Physics 130, 104703 (2009)

[7] M. H. J. Hagen, E. J. Meijer, G. C. A. M. Mooij, D. Frenkel and H. N. W. Lekkerkerker "Does C60 have a liquid phase?", Nature 365 pp. 425-426 (1993)

[8] C. Patrick Royall, and Stephen R. Williams "C60: the first one-component gel?", arXiv:1102.2959v1 (cond-mat.soft) 15 Feb 2011)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

@@ Line 1: / Line 1: @@
-{{Stub-general}}
+'''C<sub>60</sub>''', also known as ''Buckminsterfullerene'' is composed of [[carbon]] atoms.
-'''C<sub>60</sub>''', also known as ''Buckminsterfullerene''.
+{{Jmol_general|C60.pdb|C<sub>60</sub>}}
+==Models==
+====Girifalco potential====
+The Girifalco [[intermolecular pair potential]]  is given by <ref>[http://dx.doi.org/10.1021/j100181a061 L. A. Girifalco "Molecular properties of fullerene in the gas and solid phases", Journal of Physical Chemistry '''96''' pp. 858-861 (1992)]</ref> (Eq. 4):
-{|name="userboxes" id="userboxes" style="margin-left: 1em; margin-bottom: 0.5em; width: 170px; border: {{{bordercolor|#99B3FF}}} solid 1px; background-color: {{{backgroundcolor|#FFFFFF}}}; {{{extra-css|}}}" align="right"
+:<math>\Phi (r) = -\alpha \left[ \frac{1}{s(s-1)^3}+ \frac{1}{s(s+1)^3}- \frac{2}{s^4}\right] + \beta \left[ \frac{1}{s(s-1)^9}+ \frac{1}{s(s+1)^9}- \frac{2}{s^{10}}\right]</math>
-|-
-|<center>
+where
-<jmol>
-  <jmolApplet>
+:<math>s=\frac{r}{2a}</math>
-  <script>set spin X 10; spin on</script>
-  <size>200</size>
+:<math>\alpha = \frac{N^2A}{12(2a)^6}</math>
-  <color>lightgrey</color>
-    <wikiPageContents>C60.pdb</wikiPageContents>
+:<math>\beta = \frac{N^2B}{90(2a)^{12}}</math>
-  </jmolApplet>
-</jmol></center>
+where <math>N</math> is the number of atoms on each sphere, i.e. N=60.
-|-
+====Approximate non-conformal potential====
-| <center>C<sub>60</sub></center>
+The [[Approximate non-conformal potential]] ('''ANC''') for the C60 fullerene is given by (Eq 6 in <ref>[http://dx.doi.org/10.1016/j.chemphys.2017.05.014 Jesús Eloy Ramos "Effective intermolecular potential and critical point for C60 molecule", Chemical Physics '''492''' pp. 5-11 (2017)]</ref>):
-|-
-|}
+:<math>\Phi_{12}(z) =  \epsilon \left[ \frac{1-a}{(z^3/S +1 - 1/S)^{1/3} -a} \right]^{12} - 2\epsilon \left[ \frac{1-a}{(z^3/S +1 - 1/S)^{1/3} -a} \right]^{6} </math>
-[[category: Contains Jmol]]
+where
+* <math>z := r/r_m</math>
+* <math>r_m</math> = 1.0281 nm
+* <math> a </math> = 0.09574 is the hard-core diameter in units of <math>r_m</math>
+* <math> \epsilon </math> = 3297.28 K is the well depth
+* <math>S</math> = 0.4120 is a softness parameter
-===Girifalco potential===
+==Phase diagram==
-(Ref. 1)
+<ref>[http://dx.doi.org/10.1103/PhysRevLett.71.1200      Ailan Cheng, Michael L. Klein and Carlo Caccamo "Prediction of the phase diagram of rigid C60 molecules", Physical Review Letters '''71''' pp. 1200-1203 (1993)]</ref>
+<ref>[http://dx.doi.org/10.1103/PhysRevB.50.1301 L. Mederos and G. Navascués "High-temperature phase diagram of the fullerene C60"  Physical Review B '''50''' pp. 1301-1304 (1994)]</ref>
+<ref>[http://dx.doi.org/10.1063/1.479891 M. Hasegawa and K. Ohno "Monte Carlo simulation study of the high-temperature phase diagram of model C60 molecules", Journal of Chemical Physics '''111''' pp. 5955- (1999)]</ref>
+<ref>[http://dx.doi.org/10.1063/1.3081140 Pedro Orea "Phase diagrams of model C60 and C70 fullerenes from short-range attractive potentials", Journal of Chemical Physics  130, 104703 (2009)]</ref>
+==Liquid phase==
+<ref>[http://dx.doi.org/10.1038/365425a0 M. H. J. Hagen, E. J. Meijer, G. C. A. M. Mooij, D. Frenkel and H. N. W. Lekkerkerker "Does C60 have a liquid phase?", Nature '''365''' pp. 425-426 (1993)]</ref>
+==Gel phase==
+Simulations of the Girifalco potential indicate a possible [[Gels|gel]] composed solely  of C<sub>60</sub> molecules <ref>[http://arxiv.org/abs/1102.2959 C. Patrick Royall, and Stephen R. Williams "C60: the first one-component gel?", 	arXiv:1102.2959v1 (cond-mat.soft) 15 Feb 2011)]</ref>
 ==References==
-#[http://dx.doi.org/10.1021/j100181a061 L. A. Girifalco "Molecular properties of fullerene in the gas and solid phases", Journal of Physical Chemistry '''96''' pp. 858-861 (1992)]
+<references/>
+'''Related reading'''
+*[http://dx.doi.org/10.1103/PhysRevB.51.3387 C. Caccamo "Modified-hypernetted-chain determination of the phase diagram of rigid C60 molecules", Physical Review B '''51''' pp. 3387-3390 (1995)]
+*[http://dx.doi.org/10.1103/PhysRevE.54.3928 M. Hasegawa and K. Ohno "Density functional theory for the phase diagram of rigid C60 molecules", Physical Review E '''54''' pp. 3928-3932 (1996)]
+*[http://dx.doi.org/10.1063/1.473192 C. Caccamo, D. Costa, and A. Fucile "A Gibbs ensemble Monte Carlo study of phase coexistence in model C60", Journal of Chemical Physics '''106''' pp.  255- (1997)]
+*[http://dx.doi.org/10.1080/00268970902881979 M. Bahaa Khedr,  S. M. Osman and M.S. Al Busaidi "Surface tension, shear viscosity and isothermal compressibility of liquid C60 along the liquid-vapour coexistence", Molecular Physics '''107''' pp. 1355-1366 (2009)]
+*[http://dx.doi.org/10.1063/1.4866451  Minkyu Kim, Jaeeon Chang and Stanley I. Sandler "Monte Carlo simulations for the free energies of C60 and C70 fullerene crystals by acceptance ratio method and expanded ensemble method", Journal of Chemical Physics '''140''' 084110 (2014)]
+*[http://dx.doi.org/10.1063/1.4932591  D. M. Edmunds, P. Tangney, D. D. Vvedensky and W. M. C. Foulkes "Free-energy coarse-grained potential for C60", Journal of Chemical Physics '''143''' 164509 (2015)]
 [[category: models]]
+[[category: Contains Jmol]]