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Entropy

2012-01-04T06:00:04Z

70.135.118.126:

:'' "Energy has to do with possibilities. Entropy has to do with the probabilities of those possibilities happening. It takes energy and performs a further epistemological step." ''
::::: '''[[Constantino Tsallis]]''' <ref>http://www.mlahanas.de/Greeks/new/Tsallis.htm</ref>
'''Entropy''' was first described by [[Rudolf Julius Emanuel Clausius]] in 1865 <ref>[http://dx.doi.org/10.1002/andp.18652010702 R. Clausius "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie", Annalen der Physik und Chemie '''125''' pp. 353-400 (1865)]</ref>. The [[statistical mechanics | statistical mechanical]] desciption is due to [[Ludwig Eduard Boltzmann]] (Ref. ?).
==Classical thermodynamics==
In [[classical thermodynamics]] one has the entropy, <math>S</math>,
:<math>{\mathrm d} S = \frac{\delta Q_{\mathrm {reversible}}} {T} </math>

where <math>Q</math> is the [[heat]] and <math>T</math> is the [[temperature]].
==Statistical mechanics==
In [[statistical mechanics]] entropy is defined by

:<math>\left. S \right. = -k_B \sum_m p_m \ln p_m</math>

where <math>k_B</math> is the [[Boltzmann constant]], ''m'' is the index for the [[microstate |microstates]], and <math>p_m</math>
is the probability that microstate ''m'' is occupied.
In the [[microcanonical ensemble]] this gives:

:<math>\left.S\right. = k_B \ln \Omega</math>

where <math>\Omega</math> (sometimes written as <math>W</math>)
is the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system.
This equation provides a link between [[Classical thermodynamics | classical thermodynamics]] and
[[Statistical mechanics | statistical mechanics]]

==Arrow of time==
Articles:
*[http://dx.doi.org/10.1119/1.1942052 T. Gold "The Arrow of Time", American Journal of Physics '''30''' pp. 403-410 (1962)]
*[http://dx.doi.org/10.1063/1.881363 Joel L. Lebowitz "Boltzmann's Entropy and Time's Arrow", Physics Today '''46''' pp. 32-38 (1993)]
*[http://dx.doi.org/10.1023/A:1023715732166 Milan M. Ćirković "The Thermodynamical Arrow of Time: Reinterpreting the Boltzmann–Schuetz Argument", Foundations of Physics '''33''' pp. 467-490 (2003)]
Books:
* Steven F. Savitt (Ed.) "Time's Arrows Today: Recent Physical and Philosophical Work on the Direction of Time", Cambridge University Press (1997) ISBN 0521599458
* Michael C. Mackey "Time's Arrow: The Origins of Thermodynamic Behavior" (1992) ISBN 0486432432
* Huw Price "Time's Arrow and Archimedes' Point New Directions for the Physics of Time" Oxford University Press (1997) ISBN 978-0-19-511798-1

==See also:==
*[[Entropy of a glass]]
*[[H-theorem]]
*[[Non-extensive thermodynamics]]
*[[Shannon entropy]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1119/1.1990592 Karl K. Darrow "The Concept of Entropy", American Journal of Physics '''12''' pp. 183-196 (1944)]
*[http://dx.doi.org/10.1119/1.1971557 E. T. Jaynes "Gibbs vs Boltzmann Entropies", American Journal of Physics '''33''' pp. 391-398 (1965)]
*[http://dx.doi.org/10.1119/1.1287353 Daniel F. Styer "Insight into entropy", American Journal of Physics '''86''' pp. 1090-1096 (2000)]
*[http://www.ucl.ac.uk/~ucesjph/reality/entropy/text.html S. F. Gull "Some Misconceptions about Entropy" in Brian Buck and Vincent A. MacAulay (Eds.) "Maximum Entropy in Action", Oxford Science Publications (1991)]
*[http://dx.doi.org/10.2174/1874396X00802010007 Efstathios E. Michaelides "Entropy, Order and Disorder", The Open Thermodynamics Journal '''2''' pp. (2008)]
*Ya. G. Sinai, "On the Concept of Entropy of a Dynamical System," Doklady Akademii Nauk SSSR '''124''' pp. 768-771 (1959)
*[http://dx.doi.org/10.1063/1.1670348 William G. Hoover "Entropy for Small Classical Crystals", Journal of Chemical Physics '''49''' pp. 1981-1982 (1968)]
*Arieh Ben-Naim "Entropy Demystified: The Second Law Reduced to Plain Common Sense", World Scientific (2008) ISBN 978-9812832252
*Arieh Ben-Naim "Farewell to Entropy: Statistical Thermodynamics Based on Information", World Scientific (2008) ISBN 978-981-270-707-9
==External links==
*[http://www.mdpi.com/journal/entropy entropy] an international and interdisciplinary Open Access journal of entropy and information studies.
*[http://dx.doi.org/10.4249/scholarpedia.3448 Joel L. Lebowitz "Time's arrow and Boltzmann's entropy", Scholarpedia, 3(4):3448 (2008)]
[[category: statistical mechanics]]
[[category: Classical thermodynamics]]
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DL POLY

2012-01-04T05:59:46Z

70.135.118.126:

[http://www.cse.clrc.ac.uk/ccg/software/DL_POLY/index.shtml DL_POLY] is a general purpose serial and parallel [[molecular dynamics]] simulation package developed at Daresbury Laboratory by W. Smith, T.R. Forester and I.T. Todorov <ref>[http://dx.doi.org/10.1016/S0263-7855(96)00043-4 W. Smith and T. R. Forester "DL_POLY_2.0: A general-purpose parallel molecular dynamics simulation package", Journal of Molecular Graphics '''14''' pp. 136-141 (1996)]</ref>
<ref>[http://dx.doi.org/10.1080/08927020290018769 W. Smith, C. W. Yong and P. M. Rodger "DL_POLY: application to molecular simulation", Molecular Simulation '''28''' pp. 385-471 (2002)]</ref>
==Units==
DL_POLY employs an interesting set of units which have molecular relevance <ref> source: DL_POLY User Manual (v. 2.20) § 1.3.10</ref>.:
{| border="1"
|-
| physical quantity || symbol || unit value
|-
| time || <math>t_0 </math> || <math>1\times10^{-12} </math> seconds (picoseconds)
|-
| length || <math>l_0 </math>|| <math>1\times10^{-10} </math> metres (Angstroms)
|-
| mass || <math>m_0 </math>|| <math>1.66054\times10^{-27} </math> kilograms ([[Atomic mass units|amu]])
|-
| charge || <math>q_0 </math>|| <math>1.60218\times10^{-19} </math> Coulombs (electron charge)
|-
| energy || <math>E_0=m_0(l_0/t_0)^2 </math>|| <math>1.66054\times10^{-23} </math> Joules = 10 J mol<math>^{-1}</math>
|-
| [[pressure]] || <math>p_0=E_0/l_0^3 </math>|| <math>1.66054\times10^{7} </math> Pascal = 166.054 bar
|-
| [[Planck constant]] || <math>\hbar</math>|| <math>6.35078 E_0 t_0 </math>
|-
| [[Boltzmann constant]] || <math>k_B</math>|| <math>0.83145 E_0/K </math>
|}
==Force field==
The [[force fields | force field]] used in DL_POLY consists (or can consist) of the following components <ref> source: DL_POLY User Manual (v. 2.20) § 4.1.3</ref>:
*Chemical bond potentials:
**[[Harmonic bond potential]]
**[[Morse potential]]
**[[Lennard-Jones model | Lennard-Jones 12-6]]
**[[Restraint bond potential]]
**[[Quartic bond potential]]
**[[Buckingham potential]]
**[[FENE potential]]
*Angle potentials:
**[[Harmonic angle potential]]
**[[Quartic angle potential]]
**[[Truncated harmonic angle potential]]
**[[Screened harmonic angle potential]]
**[[Screened Vessal angle potential]]
**[[Truncated Vessal angle potential]]
**[[Harmonic cosine angle potential]]
**[[Cosine angle potential]]
**[[MM stretch-bend angle potential]]
**[[COMPASS stretch-stretch]]
**[[COMPASS stretch-bend]]
**[[COMPASS all terms]]
*Dihedral angle potentials:
**[[Cosine dihedral angle potential]]
**[[Harmonic dihedral angle potential]]
**[[Harmonic cosine dihedral angle potential]]
**[[Triple cosine dihedral angle potential]]
**[[Ryckaert-Bellemans dihedral angle potential]]
***[[Flourinated Ryckaert-Bellemans dihedral angle potential]]
**[[OPLS | OPLS dihedral angle potential]]
*Inversion angle potentials:
**[[Harmonic inversion angle potential]]
**[[Harmonic inversion angle potential]]
**[[Planar inversion angle potential]]
*Tethering potentials:
*Non-bonded potentials
*Three-body potentials
*Four-body potentials
*Metal potentials
**[[Embedded atom model]]
**[[Finnis-Sinclair]]
**[[Sutton-Chen]]
**[[Gupta potential]]
*External fields
==Constraint algorithms==
DL POLY can use either the [[SHAKE]] or the [[RATTLE]] algorithms as well as [[Q-SHAKE]] <ref> source: DL_POLY User Manual (v. 2.20) § 2.5.2</ref>.
==Versions of DL_POLY==
The current version of DL_POLY is DL_POLY_4
====Previos versions====
;DL_POLY_2
DL_POLY_2 was designed for simulations of up to 30,000 atoms and on parallel computers using up to 100 processors.
;DL_POLY_3
DL_POLY_3 was designed for simulations of order 100,000 to 1,000,000 atoms running on up to 1000 processors.
*DL_POLY_3 does not handle rigid body molecules.
====Other versions====
;DL_MULTI
A DL_POLY package to simulate rigid molecules with multipoles.
*[http://dx.doi.org/10.1080/00268970802175308 M. Leslie "DL_MULTI—A molecular dynamics program to use distributed multipole electrostatic models to simulate the dynamics of organic crystals", Molecular Physics '''106''' pp. 1567-1578 (2008)]

==Visualising DL_POLY output==
The visualisation program [[VMD]] is capable of displaying the HISTORY trajectory file.
==References==
<references/>
==External links==
*[http://www.cse.scitech.ac.uk/ccg/software/DL_POLY/MANUALS/USRMAN2.20.pdf DL_POLY_2.20 User Manual (PDF)]
*[http://www.cse.scitech.ac.uk/ccg/software/DL_POLY/MANUALS/USRMAN3.10.pdf DL_POLY_3.10 User Manual (PDF)]

[[Category: Materials modelling and computer simulation codes]]
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Critical points

2012-01-04T05:59:36Z

70.135.118.126:

[[Image:press_temp.png|thumb|right]]
The '''critical point''', discovered in 1822 by Charles Cagniard de la Tour <ref>Charles Cagniard de la Tour "Exposé de quelques résultats obtenu par l'action combinée de la chaleur et de la compression sur certains liquides, tels que l'eau, l'alcool, l'éther sulfurique et l'essence de pétrole rectifiée", Annales de chimie et de physique '''21''' pp. 127-132 (1822)</ref><ref>[http://dx.doi.org/10.1590/S1806-11172009000200015 Bertrand Berche, Malte Henkel, and Ralph Kenna "Critical phenomena: 150 years since Cagniard de la Tour", Revista Brasileira de Ensino de Física '''31''' pp.2602.1-2602.4 (2009)] (in English [http://arxiv.org/abs/0905.1886v1 arXiv:0905.1886v1])</ref> , is a point found at the end of the liquid-vapour coexistence curve (the red point shown on the [[pressure-temperature]] plot on the right). At this point the [[temperature]] is known as the ''critical temperature'' <math>(T_c)</math>
and the [[pressure]] is known as the ''critical pressure'' <math>(P_c)</math>.
For an interesting discourse on the "discovery" of the liquid-vapour critical point, the Bakerian Lecture of [[Thomas Andrews]]
makes good reading <ref>[http://links.jstor.org/sici?sici=0261-0523%281869%29159%3C575%3ATBLOTC%3E2.0.CO%3B2-0 Thomas Andrews "The Bakerian Lecture: On the Continuity of the Gaseous and Liquid States of Matter", Philosophical Transactions of the Royal Society of London '''159''' pp. 575-590 (1869)]</ref>. Critical points are singularities in the [[partition function]].
In the critical point vicinity (Ref. <ref>[http://dx.doi.org/10.1080/00268978300102111 G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V", Molecular Physics '''49''' pp. 1495-1504 (1983)]</ref> Eq. 17a)

:<math> \left.\frac{\partial P}{\partial n}\right\vert_{T} \simeq 0</math>

and

:<math>n \int_0^{\infty} c(r) ~4 \pi r^2 ~{\rm d}r \simeq 1</math>

For a review of the critical region see the work of Michael E. Fisher <ref>[http://dx.doi.org/10.1063/1.1704197 Michael E. Fisher "Correlation Functions and the Critical Region of Simple Fluids", Journal of Mathematical Physics '''5''' pp. 944-962 (1964)]</ref>
<blockquote>
"... Turning now to the question of specific heats, it has long been known
that real gases exhibit a large ``anomalous" specific-heat maximum
above <math>T_c</math> which lies near the critical isochore and which is not expected on classical theory..."
</blockquote>
also
<blockquote>
"... measurements (Ref. <ref>[http://dx.doi.org/10.1016/S0031-8914(58)80093-2 A. Michels, J.M. Levelt and G.J. Wolkers "Thermodynamic properties of argon at temperatures between 0°C and −140°C and at densities up to 640 amagat (pressures up to 1050 atm.)", Physica '''24''' pp. 769-794 (1958)]</ref> ) of <math>C_V(T)</math> for argon along the critical isochore suggest strongly that
<math>C_V(T) \rightarrow \infty ~{\rm as} ~ T \rightarrow T_c \pm</math>. Such a result is again inconsistent with classical theory."
</blockquote>
Thus in the vicinity of the liquid-vapour critical point, both the [[Compressibility | isothermal compressibility]]
and the [[heat capacity]] at constant pressure diverge to infinity.
==Liquid-liquid critical point==
==Solid-liquid critical point==
It is widely held that there is no solid-liquid critical point. The reasoning behind this was given on the grounds of symmetry by Landau and Lifshitz
<ref>L. D. Landau and E. M. Lifshitz, "Statistical Physics" (Course of Theoretical Physics, Volume 5) 3rd Edition Part 1, Chapter XIV, Pergamon Press (1980) § 83 p. 258</ref>. However, recent work using the [[Z1 and Z2 potentials |Z2 potential]] suggests that this may not be the last word on the subject.
<ref>[http://dx.doi.org/10.1063/1.3213616 Måns Elenius and Mikhail Dzugutov "Evidence for a liquid-solid critical point in a simple monatomic system", Journal of Chemical Physics 131, 104502 (2009)]</ref>.

==Tricritical points==
*[http://dx.doi.org/10.1103/PhysRevLett.24.715 Robert B. Griffiths "Thermodynamics Near the Two-Fluid Critical Mixing Point in He3 - He4", Physical Review Letters '''24''' 715-717 (1970)]
*[http://dx.doi.org/10.1063/1.451007 Lech Longa "On the tricritical point of the nematic–smectic A phase transition in liquid crystals", Journal of Chemical Physics '''85''' pp. 2974-2985 (1986)]
==Critical exponents==
:''Main article: [[Critical exponents]]''
==Yang-Yang anomaly==
:''Main article: [[Yang-Yang anomaly]]''
==See also==
*[[Binder cumulant]]
*[[Law of corresponding states]]
==References==
<references/>
'''Related reading'''
* M. I. Bagatskii and A. V. Voronel and B. G. Gusak "", Journal of Experimental and Theoretical Physics '''16''' pp. 517- (1963)
* [http://dx.doi.org/10.1103/PhysRevA.2.1047 Robert B. Griffiths and John C. Wheeler "Critical Points in Multicomponent Systems", Physical Review A '''2''' 1047 - 1064 (1970)]
* [http://dx.doi.org/10.1103/RevModPhys.46.597 Michael E. Fisher "The renormalization group in the theory of critical behavior", Reviews of Modern Physics '''46''' pp. 597 - 616 (1974)]
* [http://dx.doi.org/10.1146/annurev.pc.37.100186.001201 J. V. Sengers and J. M. H. Levelt Sengers "Thermodynamic Behavior of Fluids Near the Critical Point", Annual Review of Physical Chemistry '''37''' pp. 189-222 (1986)]
* [http://dx.doi.org/10.1103/PhysRevLett.93.015701 Kamakshi Jagannathan and Arun Yethiraj "Molecular Dynamics Simulations of a Fluid near Its Critical Point", Physical Review Letters '''93''' 015701 (2004)]
*[http://dx.doi.org/10.1080/00268976.2010.495734 Kurt Binder "Computer simulations of critical phenomena and phase behaviour of fluids", Molecular Physics '''108''' pp. 1797-1815 (2010)]
;Books
* H. Eugene Stanley "Introduction to Phase Transitions and Critical Phenomena", Oxford University Press (1971) ISBN 9780195053166
* Cyril Domb "The Critical Point: A Historical Introduction To The Modern Theory Of Critical Phenomena", Taylor and Francis (1996) ISBN 9780748404353

[[category: statistical mechanics]]
[[category:classical thermodynamics]]
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Critical exponents

2012-01-04T05:59:27Z

70.135.118.126:

'''Critical exponents'''. Groups of critical exponents form [[universality classes]].
==Reduced distance: <math>\epsilon</math>==
<math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e.

:<math>\epsilon = \left| 1 -\frac{T}{T_c}\right|</math>

Note that this implies a certain symmetry when the [[Critical points|critical point]] is approached from either 'above' or 'below', which is not necessarily the case.
==Heat capacity exponent: <math>\alpha</math>==
The isochoric [[heat capacity]] is given by <math>C_v</math>

:<math>\left. C_v\right.=C_0 \epsilon^{-\alpha}</math>

Theoretically one has <math>\alpha = 0.1096(5)</math><ref name="Campostrini2002">[http://dx.doi.org/10.1103/PhysRevE.65.066127 Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E '''65''' 066127 (2002)]</ref> for the three dimensional [[Ising model]], and <math>\alpha = -0.0146(8)</math><ref name="Campostrini2001" >[http://dx.doi.org/10.1103/PhysRevB.63.214503 Massimo Campostrini, Martin Hasenbusch, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "Critical behavior of the three-dimensional XY universality class" Physical Review B '''63''' 214503 (2001)]</ref> for the three-dimensional XY [[Universality classes |universality class]].
Experimentally <math>\alpha = 0.1105^{+0.025}_{-0.027}</math><ref>[http://dx.doi.org/10.1103/PhysRevE.59.1795 A. Haupt and J. Straub "Evaluation of the isochoric heat capacity measurements at the critical isochore of SF6 performed during the German Spacelab Mission D-2", Physical Review E '''59''' pp. 1795-1802 (1999)]</ref>.

==Magnetic order parameter exponent: <math>\beta</math>==
The magnetic order parameter, <math>m</math> is given by

:<math>\left. m\right. = m_0 \epsilon^\beta</math>

Theoretically one has <math>\beta =0.32653(10)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising |three dimensional Ising model]], and <math>\beta = 0.3485(2)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
==Susceptibility exponent: <math>\gamma</math>==
[[Susceptibility]]

:<math>\left. \chi \right. = \chi_0 \epsilon^{-\gamma}</math>

Theoretically one has <math>\gamma = 1.2373(2)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising |three dimensional Ising model]], and <math>\gamma = 1.3177(5)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
==Correlation length==

:<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math>

Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising |three dimensional Ising model]], and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
==Inequalities==
====Fisher inequality====
The Fisher inequality (Eq. 5 <ref>[http://dx.doi.org/10.1103/PhysRev.180.594 Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review '''180''' pp. 594-600 (1969)]</ref>)

:<math>\gamma \le (2-\eta) \nu</math>
====Griffiths inequality====
The Griffiths inequality (Eq. 3 <ref>[http://dx.doi.org/10.1103/PhysRevLett.14.623 Robert B. Griffiths "Thermodynamic Inequality Near the Critical Point for Ferromagnets and Fluids", Physical Review Letters '''14''' 623-624 (1965)]</ref>):

:<math>(1+\delta)\beta \ge 2-\alpha'</math>
====Josephson inequality====
The Josephson inequality <ref>[http://dx.doi.org/10.1088/0370-1328/92/2/301 B. D. Josephson "Inequality for the specific heat: I. Derivation", Proceedings of the Physical Society '''92''' pp. 269-275 (1967)]</ref><ref>[http://dx.doi.org/10.1088/0370-1328/92/2/302 B. D. Josephson "Inequality for the specific heat: II. Application to critical phenomena", Proceedings of the Physical Society '''92''' pp. 276-284 (1967)]</ref><ref>[http://dx.doi.org/10.1007/BF01008478 Alan D. Sokal "Rigorous proof of the high-temperature Josephson inequality for critical exponents", Journal of Statistical Physics '''25''' pp. 51-56 (1981)]</ref>

:<math>d\nu \ge 2-\alpha</math>
====Liberman inequality====
<ref>[http://dx.doi.org/10.1063/1.1726488 David A. Liberman "Another Relation Between Thermodynamic Functions Near the Critical Point of a Simple Fluid", Journal of Chemical Physics '''44''' 419-420 (1966)]</ref>
====Rushbrooke inequality====
The Rushbrooke inequality (Eq. 2 <ref>[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)]</ref>), based on the work of Essam and Fisher (Eq. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)]</ref>) is given by

:<math>\alpha' + 2\beta + \gamma' \ge 2</math>.

Using the above-mentioned values<ref name="Campostrini2002"> </ref> one has:

:<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math>
====Widom inequality====
The Widom inequality <ref>[http://dx.doi.org/10.1063/1.1726135 B. Widom "Degree of the Critical Isotherm", Journal of Chemical Physics '''41''' pp. 1633-1634 (1964)]</ref>

:<math>\gamma' \ge \beta(\delta -1)</math>
==Hyperscaling==
==Gamma divergence==
When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form

:<math>\left. \right. \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}</math>

where <math>\kappa_T</math> is the [[Compressibility#Isothermal compressibility | isothermal compressibility]]. <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state#Critical exponents | Van der Waals equation of state]], and is usually 1.2 to 1.3.

==Epsilon divergence==
When approaching the critical point along the critical isotherm the divergence is of the form

:<math>\left. \right. \kappa_T \sim (p-p_c)^{-\epsilon}</math>

where <math>\epsilon</math> is 2/3 for the [[Van der Waals equation of state]], and is usually 0.75 to 0.8.
==References==
<references/>
[[category: statistical mechanics]]
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COMPASS force field

2012-01-04T05:59:08Z

70.135.118.126:

'''COMPASS''' ('''C'''ondensed-phase '''O'''ptimized '''M'''olecular '''P'''otentials for '''A'''tomistic '''S'''imulation '''S'''tudies) is a member of the consistent family of [[force fields]] ([[CFF91]], [[PCFF]], [[CFF]] and COMPASS), which are closely related second-generation force fields. They were parameterized against a wide range of experimental observables for organic compounds containing [[hydrogen |H]], [[carbon |C]], [[nitrogen |N]], [[oxygen |O]], [[Sulfur |S]], [[Phosphorus |P]], halogen atoms and ions, alkali metal cations, and several biochemically important divalent metal cations. PCFF is based on CFF91, extended so as to have a broad coverage of organic polymers, (inorganic) metals, and [[zeolites]]. COMPASS is the first force field that has been parameterized and validated using condensed phase properties in addition to various and empirical data for molecules in isolation. Consequently, this forcefield enables accurate and simultaneous prediction of structural, conformational, vibrational, and thermo-physical properties for a broad range of molecules in isolation and in condensed phases.
==Functional form==
The COMPASS force field (Ref. 1 Eq. 1) consists of terms for bonds (<math>b</math>), angles (<math>\theta</math>), dihedrals (<math>\phi</math>), out-of-plane angles (<math>\chi</math>) as well as cross-terms, and two non-bonded functions, a [[Coulomb's law |Coulombic function]] for electrostatic interactions and a [[9-6 Lennard-Jones potential]] for van der Waals interactions.

:<math>E_{\mathrm{total}} = E_b + E_{\theta} + E_{\phi} + E_{\chi} + E_{b,b'} + E_{b,\theta} + E_{b,\phi} + E_{\theta,\phi} +
E_{\theta,\theta'} + E_{\theta,\theta',\phi} + E_{q} + E_{\mathrm{vdW}}</math>

where

:<math>E_b = \sum_b \left[ k_2 (b-b_0)^2 + k_3 (b-b_0)^3 + k_4 (b-b_0)^4 \right]</math>

:<math>E_{\theta} = \sum_{\theta} \left[ k_2 (\theta-\theta_0)^2 + k_3 (\theta-\theta_0)^3 + k_4 (\theta-\theta_0)^4 \right]</math>

:<math>E_{\phi} = \sum_{\phi} \left[ k_1(1-\cos \phi) + k_2(1-\cos 2\phi) + k_3(1-\cos 3\phi) \right]</math>

:<math>E_{\chi} = \sum_{\chi} k_2 \chi^2</math>

:<math>E_{b,b'} = \sum_{b,b'} k(b-b_0)(b'-b_0')</math>

:<math>E_{b,\theta} = \sum_{b,\theta} k(b-b_0)(\theta-\theta_0)</math>

:<math>E_{b,\phi} = \sum_{b,\phi} (b-b_0) \left[k_1 cos \phi + k_2 cos 2\phi + k_3 cos 3\phi\right]</math>

:<math>E_{\theta,\phi} = \sum_{\theta,\phi} (\theta-\theta_0) \left[k_1 cos \phi + k_2 cos 2\phi + k_3 cos 3\phi\right]</math>

:<math>E_{\theta,\theta'} = \sum_{\theta,\theta'} k(\theta-\theta_0)(\theta'-\theta_0')</math>

:<math>E_{\theta,\theta',\phi} = \sum_{\theta,\theta',\phi} k(\theta-\theta_0)(\theta'-\theta_0')\cos \phi</math>

:<math>E_{q}=\sum_{ij} \frac{q_i q_j}{r_{ij}}</math>

and

:<math>E_{\mathrm{vdW}}= \sum_{ij} \epsilon_{ij} \left[ 2 \left(\frac{r_{ij}^0}{r_{ij}} \right)^9 -3 \left(\frac{r_{ij}^0}{r_{ij}} \right)^6\right]</math>

with the following [[combining rules]] (Ref. 1 Eqs. 2 and 3):

:<math>r_{ij}^0 = \left( \frac{ (r_i^0)^6 + (r_j^0)^6 }{2} \right)^{1/6}</math>

and

:<math>\epsilon_{ij} = 2 \sqrt{\epsilon_i \cdot \epsilon_j} \left( \frac{ (r_i^0)^3 \cdot (r_j^0)^3 }{ (r_i^0)^6 \cdot (r_j^0)^6 } \right)</math>

==Parameters==
==References==
#[http://dx.doi.org/10.1021/jp980939v H. Sun "COMPASS: An ab Initio Force-Field Optimized for Condensed-Phase Applications - Overview with Details on Alkane and Benzene Compounds", Journal of Physical Chemistry B '''102''' pp. 7338–7364 (1998)]
#[http://dx.doi.org/10.1016/S1089-3156(98)00042-7 H. Sun, P. Ren and J. R. Fried "The COMPASS force field: parameterization and validation for phosphazenes", Computational and Theoretical Polymer Science '''8''' pp. 229-246 (1998)]
#[http://dx.doi.org/10.1021/jp991786u S. W. Bunte and H. Sun "Molecular Modeling of Energetic Materials: The Parameterization and Validation of Nitrate Esters in the COMPASS Force Field", Journal of Physical Chemistry B '''104''' pp. 2477-2489 (2000)]
#[http://dx.doi.org/10.1021/jp992913p Jie Yang, Yi Ren, An-min Tian and Huai Sun "COMPASS Force Field for 14 Inorganic Molecules, He, Ne, Ar, Kr, Xe, H2, O2, N2, NO, CO, CO2, NO2, CS2, and SO2, in Liquid Phases", Journal of Physical Chemistry B '''104''' pp. 4951-4957 (2000)]
#[http://dx.doi.org/10.1002/jcc.10316 Michael J. McQuaid, Huai Sun and David Rigby "Development and validation of COMPASS force field parameters for molecules with aliphatic azide chains", Journal of Computational Chemistry '''25''' pp. 61-71 (2004)]
[[category: force fields]]
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Combining rules

2012-01-04T05:58:59Z

70.135.118.126:

The '''combining rules''' are geometric expressions designed to provide the interaction energy between two dissimilar non-bonded atoms (here labelled <math>i</math> and <math>j</math>). Most of the rules are designed to be used with a specific [[Idealised models| interaction potential]] in mind. (''See also'' [[Mixing rules]]).
==Böhm-Ahlrichs==
<ref>[http://dx.doi.org/10.1063/1.444057 Hans‐Joachim Böhm and Reinhart Ahlrichs "A study of short‐range repulsions", Journal of Chemical Physics '''77''' pp. 2028- (1982)]</ref>
==Diaz Peña-Pando-Renuncio==
<ref>[http://dx.doi.org/10.1063/1.442726 M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. I. Rules based on approximations for the long-range dispersion energy", Journal of Chemical Physics '''76''' pp. 325- (1982)]</ref>
<ref>[http://dx.doi.org/10.1063/1.442727 M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. II. Rules based on approximations for the long-range dispersion energy and an atomic distortion model for the repulsive interactions", Journal of Chemical Physics '''76''' pp. 333- (1982)]</ref>
==Fender-Halsey==
The Fender-Halsey combining rule for the [[Lennard-Jones model]] is given by <ref>[http://dx.doi.org/10.1063/1.1701284 B. E. F. Fender and G. D. Halsey, Jr. "Second Virial Coefficients of Argon, Krypton, and Argon-Krypton Mixtures at Low Temperatures", Journal of Chemical Physics '''36''' pp. 1881-1888 (1962)]</ref>
:<math>\epsilon_{ij} = \frac{2 \epsilon_i \epsilon_j}{\epsilon_i + \epsilon_j}</math>
==Gilbert-Smith==
The Gilbert-Smith rules for the [[Born-Huggins-Meyer potential]]<ref>[http://dx.doi.org/10.1063/1.1670463 T. L. Gilbert "Soft‐Sphere Model for Closed‐Shell Atoms and Ions", Journal of Chemical Physics '''49''' pp. 2640- (1968)]</ref><ref>[http://dx.doi.org/10.1063/1.431848 T. L. Gilbert, O. C. Simpson, and M. A. Williamson "Relation between charge and force parameters of closed‐shell atoms and ions", Journal of Chemical Physics '''63''' pp. 4061- (1975)]</ref><ref>[http://dx.doi.org/10.1103/PhysRevA.5.1708 Felix T. Smith "Atomic Distortion and the Combining Rule for Repulsive Potentials", Physical Review A '''5''' pp. 1708-1713 (1972)]</ref>.
==Good-Hope rule==
The Good-Hope rule for [[Mie potential |Mie]]–[[Lennard-Jones model |Lennard‐Jones]] or [[Buckingham potential]]s <ref>[http://dx.doi.org/10.1063/1.1674022 Robert J. Good and Christopher J. Hope "New Combining Rule for Intermolecular Distances in Intermolecular Potential Functions", Journal of Chemical Physics '''53''' pp. 540- (1970)]</ref> is given by (Eq. 2):

:<math>\sigma_{ij} = \sqrt{\sigma_{ii} \sigma_{jj}}</math>
==Hudson and McCoubrey==
<ref>[http://dx.doi.org/10.1039/TF9605600761 G. H. Hudson and J. C. McCoubrey "Intermolecular forces between unlike molecules. A more complete form of the combining rules", Transactions of the Faraday Society '''56''' pp. 761-766 (1960)]</ref>
==Kong rules==
The Kong rules for the [[Lennard-Jones model]] are given by (Table I in
<ref>[http://dx.doi.org/10.1063/1.1680358 Chang Lyoul Kong "Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential", Journal of Chemical Physics '''59''' pp. 2464-2467 (1973)]</ref>):

:<math>\epsilon_{ij}\sigma_{ij}^{6}=\left(\epsilon_{ii}\sigma_{ii}^{6}\epsilon_{jj}\sigma_{jj}^{6}\right)^{\frac{1}{2}}</math>

:<math> \epsilon_{ij}\sigma_{ij}^{12} = \frac{\epsilon_{ii}\sigma_{ii}^{12}}{2^{13}}\left[ 1+\left( \frac{\epsilon_{jj}\sigma_{jj}^{12}}{\epsilon_{ii}\sigma_{ii}^{12}} \right)^{\frac{1}{13}}\right]^{13} </math>

==Lorentz-Berthelot rules==
The Lorentz rule is given by <ref>[http://dx.doi.org/10.1002/andp.18812480110 H. A. Lorentz "Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase", Annalen der Physik '''12''' pp. 127-136 (1881)]</ref>
:<math>\sigma_{ij} = \frac{\sigma_{ii} + \sigma_{jj}}{2}</math>

which is only really valid for the [[hard sphere model]].

The Berthelot rule is given by <ref>[http://visualiseur.bnf.fr/Document/CadresPage?O=NUMM-3082&I=1703 Daniel Berthelot "Sur le mélange des gaz", Comptes rendus hebdomadaires des séances de l’Académie des Sciences, '''126''' pp. 1703-1855 (1898)]</ref>

:<math>\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}</math>

These rules are simple and widely used, but are not without their failings <ref>[http://dx.doi.org/10.1080/00268970010020041 Jérôme Delhommelle; Philippe Millié "Inadequacy of the Lorentz-Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation", Molecular Physics '''99''' pp. 619-625 (2001)]</ref>
<ref>[http://dx.doi.org/10.1080/00268970802471137 Dezso Boda and Douglas Henderson "The effects of deviations from Lorentz-Berthelot rules on the properties of a simple mixture", Molecular Physics '''106''' pp. 2367-2370 (2008)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1610435 W. Song, P. J. Rossky, and M. Maroncelli "Modeling alkane+perfluoroalkane interactions using all-atom potentials: Failure of the usual combining rules", Journal of Chemical Physics '''119''' pp. 9145- (2003)]</ref>.

==Mason-Rice rule==
The Mason-Rice rule for the [[Exp-6 potential]] <ref>[http://dx.doi.org/10.1063/1.1740100 Edward A. Mason and William E. Rice "The Intermolecular Potentials of Helium and Hydrogen", Journal of Chemical Physics '''22''' pp. 522- (1954)]</ref>.
==Sikora rules==
The Sikora rules for the [[Lennard-Jones model]] <ref>[http://dx.doi.org/10.1088/0022-3700/3/11/008 P. T. Sikora "Combining rules for spherically symmetric intermolecular potentials", Journal of Physics B: Atomic and Molecular Physics '''3''' pp. 1475- (1970)]</ref>.
==Tang and Toennies==
<ref>[http://dx.doi.org/10.1007/BF01384663 K. T. Tang and J. Peter Toennies "New combining rules for well parameters and shapes of the van der Waals potential of mixed rare gas systems", Zeitschrift für Physik D Atoms, Molecules and Clusters '''1''' pp. 91-101 (1986)]</ref>
==Waldman-Hagler rules==
<ref>[http://dx.doi.org/10.1002/jcc.540140909 M. Waldman and A. T. Hagler "New combining rules for rare-gas Van der-Waals parameters", Journal of Computational Chemistry '''14''' pp. 1077-1084 (1993)]</ref>
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1021/ja00046a032 Thomas A. Halgren "The representation of van der Waals (vdW) interactions in molecular mechanics force fields: potential form, combination rules, and vdW parameters", Journal of the American Chemical Society '''114''' pp. 7827-7843 (1992)]
[[category: mixtures]]
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CLAYFF force field

2012-01-04T05:58:48Z

70.135.118.126:

'''ClayFF'''<ref name="Cygan">[http://dx.doi.org/10.1021/jp0363287 Randall T. Cygan, Jian-Jie Liang, and Andrey G. Kaliniche "Molecular Models of Hydroxide, Oxyhydroxide, and Clay Phases and the Development of a General Force Field" Journal of Physical Chemistry B '''108''' pp. 1255-1266 (2004)]</ref> is a general [[Force fields |force field]] suitable for the simulation of hydrated and multicomponent mineral systems and their [[Interface |interfaces]] with aqueous [[solutions]].
With the issue of rising atmospheric concentration of the greenhouse (global warming) gas, [[carbon dioxide]] (CO2) also comes a burgeoning interest in novel repositories in which to inexpensively "bury" CO2 to reduce its atmospheric load. This issue, among others, has prompted scientists to examine various ubiquitous and inexpensive [[clays]] (for example, [[montmorrillonite]] or [[kaolinite]]) as potential CO2 repositories. But clays are heterogeneous, somewhat unstructured and molecularly complex entities (by comparison to, for example, pure salt --- [[sodium chloride]] --- crystals), and there are uncertainties in experimental methods for studying the binding and retention of other atoms, ions, and molecules (such as CO2) to hydrated ([[water]]-wettened) clays. Hence, it is important to apply theoretical [[Realistic models |molecular models]] to achieve a fundamental atomic-level understanding, interpretation, and prediction of these chemical phenomena. ClayFF is available in [[Materials modelling and computer simulation codes |molecular simulation codes]] (for example, [[MCCCS Towhee]] and [[OpenMD]]) and was developed by Sandia National Laboratories chemist, Randall Cygan, and collaborators at the University of Illinois at Urbana-Champaign. It is suitable for the simulation of hydrated and multicomponent mineral systems and their interfaces with aqueous solutions. The ClayFF approach treats most inter-atomic interactions as being non-bonded. This allows the use of the force field for a wide variety of phases and properly accounts for energy and momentum transfer between the fluid phase and the solid, while keeping the number of parameters small enough to permit modelling of relatively large and highly disordered systems such as clays.
==Functional form==
The functional form of ClayFF is given by (Eq. 1 <ref name="Cygan"> </ref>):

:<math>E_{\mathrm {total}} = E_{\mathrm {Coulombic}} + E_{\mathrm {VdW}} + E_{\mathrm {bond~stretch}} + E_{\mathrm {angle~bend}}</math>

where (Eq. 2 <ref name="Cygan"> </ref>):

:<math> E_{\mathrm {Coulombic}} = \frac{e^2}{4 \pi \epsilon_0} \sum_{i \neq j} \frac{q_i q_j}{r_{ij}}</math>

(Eq. 3 <ref name="Cygan"> </ref>):

:<math>E_{\mathrm {VdW}} = \sum_{i \neq j} D_{0,ij} \left[ \left( \frac{R_{0,ij}}{r_{ij}} \right)^{12} -2 \left( \frac{R_{0,ij}}{r_{ij}}\right)^{6}\right]</math>

(Eq. 6 <ref name="Cygan"> </ref>):

:<math>E_{\mathrm {bond~stretch}} = \sum_{\rm {bonds}} k_1\left(r_{ij}-r_0\right)^2</math>

(Eq. 7 <ref name="Cygan"> </ref>):

:<math> E_{\mathrm {angle~bend}} \sum_{\rm {angles}} k_2 \left(\theta_{ijk} -\theta_{0}\right)^2</math>
==Parameters==
==References==
<references/>
==External links==
*[http://www.sandia.gov/eesector/gs/gc/rtc.htm Dr. Randall T. Cygan]

[[category: Force fields]]
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Chemical potential

2012-01-04T05:58:34Z

70.135.118.126:

==Classical thermodynamics==
Definition:

:<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p} = \left. \frac{\partial A}{\partial N}\right\vert_{T,V}</math>

where <math>G</math> is the [[Gibbs energy function]], leading to

:<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math>

where <math>A</math> is the [[Helmholtz energy function]], <math>k_B</math>
is the [[Boltzmann constant]], <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>V</math>
is the volume.

==Statistical mechanics==
The chemical potential is the derivative of the [[Helmholtz energy function]] with respect to the
number of particles

:<math>\mu= \left. \frac{\partial A}{\partial N}\right\vert_{T,V}=\frac{\partial (-k_B T \ln Z_N)}{\partial N} = -\frac{3}{2} k_BT \ln \left(\frac{2\pi m k_BT}{h^2}\right) + \frac{\partial \ln Q_N}{\partial N}</math>
where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math>
identical particles
:<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math>
and <math>Q_N</math> is the
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral]
:<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math>

==Kirkwood charging formula==
The Kirkwood charging formula is given by <ref>[http://dx.doi.org/10.1063/1.1749657 John G. Kirkwood "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)]</ref>

:<math>\beta \mu_{\rm ex} = \rho \int_0^1 d\lambda \int \frac{\partial \beta \Phi_{12} (r,\lambda)}{\partial \lambda} {\rm g}(r,\lambda) dr</math>

where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]] and <math>{\rm g}(r)</math> is the [[Pair distribution function | pair correlation function]].
==See also==
*[[Ideal gas: Chemical potential]]
*[[Widom test-particle method]]
*[[Overlapping distribution method]]

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1119/1.17844 G. Cook and R. H. Dickerson "Understanding the chemical potential", American Journal of Physics '''63''' pp. 737-742 (1995)]
*[http://dx.doi.org/10.1007/s10955-005-8067-x T. A. Kaplan "The Chemical Potential", Journal of Statistical Physics '''122''' pp. 1237-1260 (2006)]
[[category:classical thermodynamics]]
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CFF force field

2012-01-04T05:58:18Z

70.135.118.126:

'''CFF''' ('''c'''onsistent '''f'''orce-'''f'''ield, formerly [[CFF95]]) is a member of the consistent family of force fields ([[CFF91]], [[PCFF]], CFF and [[COMPASS]]), which are closely related second-generation force fields. They were parameterized against a wide range of experimental observables for organic compounds containing [[hydrogen |H]], [[carbon |C]], [[nitrogen |N]], [[oxygen |O]], [[Sulfur |S]], [[Phosphorus |P]], halogen atoms and ions, alkali metal cations, and several biochemically important divalent metal cations. CFF was parameterized for additional functional groups beyond CFF91. It is recommended for all life sciences applications and for organic polymers such as polycarbonates and polysaccharides.
==Functional form==
==Parameters==
==References==
#[http://dx.doi.org/10.1002/jcc.540150207 J. R. Maple, M.-J. Hwang, T. P. Stockfisch, U. Dinur, M. Waldman, C. S. Ewig, A. T. Hagler "Derivation of class II force fields. I. Methodology and quantum force field for the alkyl functional group and alkane molecules", Journal of Computational Chemistry '''15''' pp. 162-182 (1994)]
#[http://dx.doi.org/10.1039/a909475j Svava Ósk Jónsdóttir and Kjeld Rasmussen "The consistent force field. Part 6: an optimized set of potential energy functions for primary amines", New Journal of Chemistry '''24''' pp. 243-247 (2000)]
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Capillary waves

2012-01-04T05:57:58Z

70.135.118.126:

==Thermal capillary waves==
Thermal '''capillary waves''' are oscillations of an [[interface]] which are thermal in origin. These take place at the molecular level, where only the contribution due to [[surface tension]] is relevant.
Capillary wave theory is a classic account of how thermal fluctuations distort an interface (Ref. 1). It starts from some [[intrinsic surface]] that is distorted. In the Monge representation, the surface is given as <math>z=h(x,y)</math>. An increase in area of the surface causes a proportional increase of energy:

:<math>
E_\mathrm{st}= \sigma \iint dx\, dy\ \sqrt{1+\left( \frac{dh}{dx} \right)^2+\left( \frac{dh}{dy} \right)^2} -1
</math>

for small values of the derivatives (surfaces not too rough):

:<math>
E_\mathrm{st} \approx \frac{\sigma}{2} \iint dx\, dy\ \left[ \left( \frac{dh}{dx} \right)^2+\left( \frac{dh}{dy} \right)^2 \right].
</math>

A [[Fourier analysis]] treatment begins by writing the intrinsic surface as an infinite sum of normal modes:

:<math>h(x,y)= \sum_\vec{q} a_\vec{q} e^{i\vec{q}\vec{r}}.</math>

Since normal modes are orthogonal, the energy is easily expressible as a sum of terms <math>\propto q^2 |a_\vec{q}|^2</math>. Each term of the sum is quadratic in the amplitude; hence [[equipartition]] holds, according to standard [[statistical mechanics | classical statistical mechanics]], and the mean energy of each mode will be <math>k_B T/2</math>. Surprisingly, this result leads to a '''divergent''' surface (the width of the interface is bound to diverge with its area) (Ref 2). This divergence is nevertheless very mild; even for displacements on the order of meters, the deviation of the surface is comparable to the size of the molecules.
Moreover, the introduction of an external field removes this divergence: the action of gravity is sufficient to keep the width fluctuation on the order
of one molecular diameter for areas larger than about 1 mm2 (Ref. 2).
The action of gravity is taken into account by integrating the potential energy density due to gravity, <math>\rho g z</math> from a reference height to the position of the surface, <math>z=h(x,y)</math>:

:<math>E_\mathrm{g}= \iint dx\, dy\, \int_0^h dz \rho g z = \frac{\rho g}{2} \int dx\, dy\, h^2.</math>

(For simplicity, one neglects the density of the gas above, which is often acceptable; otherwise, instead of the density the difference in densities appears).

Recently, a procedure has been proposed to obtain a molecular intrinsic
surface from simulation data (Ref. 3), the [[intrinsic sampling method]]. The density profiles obtained
from this surface are, in general, quite different from the usual
''mean density profiles''.

==Gravity-capillary waves==
These are ordinary waves excited in an interface, such as ripples on
a water surface. Their dispersion relation reads, for waves on the interface between two fluids of infinite depth:

:<math>
\omega^2=\frac{\rho-\rho'}{\rho+\rho'}gk+\frac{\sigma}{\rho+\rho'}k^3,</math>

where <math>\omega</math> is the angular frequency, <math>g</math> the acceleration due to gravity, <math>\sigma</math> the [[surface tension]], <math>\rho</math> and <math>\rho^'</math> the mass density of the two fluids (<math>\rho > \rho^'</math>) and <math>k</math> the is wavenumber.
===Derivation===
This is a sketch of the derivation of the general dispersion relation, see Ref. 4 for a more detailed description. The problem is unfortunately a bit complex. As Richard Feynman put it (Ref. 6):
<blockquote>
"''...[water waves], which are easily seen by everyone and which are used as an example of waves in elementary courses... are the worst possible example... they have all the complications that waves can have''"
</blockquote>
====Defining the problem====
Three contributions to the energy are involved: the [[surface tension]], gravity, and [[hydrodynamics]]. The parts due to surface tension (again the derivatives are taken to be small) and gravity are exactly as above.
The new contribution involves the [[kinetic energy]] of the fluid:

:<math>T= \frac{\rho}{2} \iint dx\, dy\, \int_{-\infty}^h dz v^2,</math>

where <math>v</math> is the module of the velocity field <math>\vec{v}</math>.
(Again, we are neglecting the flow of the gas above for simplicity.)

====Wave solutions====
Let us suppose the surface of the liquid is described by a traveling plane wave:

:<math>h(x,y,t)=\eta(t)e^{i\vec{q}\vec{r}},</math>

where <math>\eta(t)=\exp[i\omega t]</math> and <math>\vec{q}=(q_x,q_y)</math> is a two dimensional wave number vector, <math>\vec{r}=(x,y)</math> being the horizontal position. We may take <math>\vec{q}=(q,0)</math> without loss of generality:

:<math>h(x,y,t)=\eta(t)e^{i q x}.</math>

In this case it is easy to perform the integrations involved in the expressions for the energies. The
integration over <math>x</math> can taken over a period of oscillation <math>\lambda=2\pi/q</math>, then
multiplied by the number of oscillations in our very large (in principle, infinite) system: <math>L_x / \lambda</math>. The integration over <math>y</math> trivially yields <math>L_y</math>. Performing the integrations, keeping in mind that only the real part of complex numbers is to be taken as physical, one finds:

:<math>E_\mathrm{g} = \frac{A}{2} \frac{\rho g}{2} \eta^2,</math>
:<math>E_\mathrm{st} = \frac{A}{2} \frac{\sigma}{2} q^2 \eta^2,</math>

where <math>A=L_x\times L_y</math> is the area of the system.

To tackle the kinetic energy, suppose the fluid is incompressible and its flow is irrotational (often, sensible approximations) - the flow will then be [[potential flow|potential]]: <math>\vec{v}=\nabla\phi</math>, and <math>\phi</math> is a potential (scalar field) which must satisfy [[Laplace's equation]] <math>\nabla^2\phi=0</math>.
If we try try separation of variables with the potential:

:<math>\phi(x,y,z,t)=\xi(t) f(z) e^{i q x},</math>

with some function of time <math>\xi(t)</math>, and some function of vertical component (height) <math>f(z)</math>.
Laplace's equation then requires on the later

:<math>f''(z)= q^2 f(z) .</math>

This equation can be solved with the proper boundary conditions: first, <math>\vec{v}</math> must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more general relation holds, which is also well known in oceanography). Therefore

:<math>f(z) = a \, \exp(|q| z) </math>,

with some constant <math>a</math>. The less trivial condition is the proper matching between <math>\phi</math> and <math>h</math>: the potential field must correspond to a velocity field that is adjusted to the movement of the surface:

:<math>v_z (z=h) =\partial h/\partial t</math>.

(Actually, this is the linearized version of a more general expression, see below.)

This implies that:

:<math>\xi(t)=\eta(t)'</math>, and <math>f'(z=h) = 1, </math>

so that

:<math>f(z) = \exp( -|q|(h-z))/|q| </math>.

We may now find the velocity field, <math>\vec{v}=\nabla\phi</math>, which shows the well-known circles: the elements of fluid undergo circular motion in the <math>x,z</math> plane, with the circles getting smaller at deeper levels. The displacement of a fluid element is given by <math>\partial\vec{\psi}/\partial t= \vec{v}</math>, and is plotted in Figure 1.

[[Image:Wave_v_field.jpg|300px|thumb|right|Figure 1. Displacement of particle (snapshot)]]

For the kinetic energy, we need
<math>v^2=|\nabla\phi|^2</math>, which is:

:<math>v^2= (\eta')^2 e^{ -2 |q|(h-z)}, </math>

with no dependence on <math>x</math> or <math>y</math>; the other integration provides:

:<math>T= \frac{\rho A }{2|q|} ( \eta' )^2 \int_{-\infty}^h e^{ -2 |q|(h-z)} = \frac{A}{2} \frac{\rho }{2|q|} ( \eta' )^2 .
</math>

The problem is thus specified by just a potential energy involving the square of <math>\eta(t)</math> and a kinetic energy involving the square of its time derivative: a regular [[Harmonic spring approximation|harmonic oscillator]]. In particular:
:<math>E= \frac{A}{2}
\left[
\left( \rho g + {\sigma} q^2 \right) \frac{\eta^2}{2}+
\frac{\rho}{q} \frac{(\eta')^2}{2}
\right].
</math>
Identifying the oscillator's "spring constant" <math>\kappa = \rho g + {\sigma} q^2 </math>, and its "mass"
<math>m= \rho / q</math>, the oscillation frequency must be given by <math>\omega^2=\kappa/m</math>, which results in:

:<math>\omega^2=g q+\frac{\sigma}{\rho} q^3,</math>

the same dispersion as above if <math>\rho'</math> is neglected.

====Alternative derivation====

In Reference 7 the dispersion relation is derived in a somewhat different manner. The same assumptions are made regarding the fluid (it is inviscid, irrotational, and incompressible), so Laplace's Equation is to be satisfied: <math>\nabla^2\phi=0</math>, with <math>\vec{v}=\nabla\phi</math>. The boundary conditions, on the other hand, are sufficient to solve the problem.

One boundary condition is the requirement that the surface of the liquid, defined by <math>z=h(x,y;t)</math> follows the velocity field:
:<math>
\frac{\partial h}{\partial t}+v_x \frac{\partial h}{\partial x}+v_y \frac{\partial h}{\partial y}= v_z .
</math>
A simpler condition follows from linearization: <math> \partial h /\partial t =v_z </math>, as in the previous derivation. There is an additional boundary condition at the bottom of the fluid, which we take here as <math>v_z=0</math> when <math>z\rightarrow -\infty</math>.

In the same fashion as above, we seek surface wave solutions, of the form <math>h(x,y,t)=a e^{i (qx-\omega t)}</math>. We may guess a solution of the form
:<math>\phi=-i\omega h(x,y;t) f(z).</math>

This first condition implies <math>f'(z=h)=1</math>. Together with Laplace's equation, this leads to a function
:<math>f=(1/q) \exp(q(z-h)). </math>
(see Ref 8 for a discussion on when Laplace's equation admits wave solutions.)

The other surface boundary condition is a [[Bernoulli equation]], stating that the pressure just below the surface, <math>p_-</math>, must equal the [[saturation pressure]] of coexistence, minus a contribution due to the surface:
:<math>
p_-=p -
\rho\left[
\frac{\partial \phi}{\partial t}+
\frac{1}{2} v^2+ gh
\right] .
</math>
The linearized condition is
:<math>
\frac{\partial \phi}{\partial t}+gh = \frac{p-p_-}{\rho}
</math>

The connection with the curvature of the surface can be introduced by [[Young's equation]] for the pressure drop across a curved interface, whose linearized form is:
:<math>
p-p_-=\sigma\left(\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial y^2}\right),
</math>
where <math>\sigma</math> is the surface tension.
The linearized condition is finally
:<math>
\frac{\partial \phi}{\partial t}+gh = \frac{\sigma}{\rho} \left(\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial y^2}\right).
</math>

This second condition, when applied to the surface wave above, establishes that <math>f(z=h)=(g+\sigma/\rho q^2)/\omega^2</math>.

For the two conditions to hold, <math>1/q</math> must equal <math>(g+\sigma/\rho q^2)/\omega^2</math>, which is precisely the same dispersion relation as the one above.

This derivation makes clear the assumptions introduced. In particular, the linearization will only hold for smooth waves, the ones for which the wave amplitude, <math>a</math> is smaller than the wavelength. Mathematically, the limit is <math>q a \ll 1</math>. For ocean waves, this happens when waves approach the shore and the amplitude grows (in this limit, a bottom boundary condition <math>v_z (z=-H) =0</math> must be employed, and waves are not dispersive, see Ref 7.)
==References==
#[http://dx.doi.org/10.1103/PhysRevLett.15.621 F. P. Buff, R. A. Lovett, and F. H. Stillinger, Jr. "Interfacial density profile for fluids in the critical region" Physical Review Letters '''15''' pp. 621-623 (1965)]
#J. S. Rowlinson and B. Widom "Molecular Theory of Capillarity". Dover 2002 (originally: Oxford University Press 1982) ISBN 0486425444
#[http://dx.doi.org/10.1103/PhysRevLett.91.166103 E. Chacón and P. Tarazona "Intrinsic profiles beyond the capillary wave theory: A Monte Carlo study", Physical Review Letters '''91''' 166103 (2003)]
#Samuel A. Safran "Statistical thermodynamics of surfaces, interfaces, and membranes" Addison-Wesley 1994 ISBN 9780813340791
#[http://dx.doi.org/10.1103/PhysRevLett.99.196101 P. Tarazona, R. Checa, and E. Chacón "Critical Analysis of the Density Functional Theory Prediction of Enhanced Capillary Waves", Physical Review Letters '''99''' 196101 (2007)]
#R.P. Feynman, R.B. Leighton, and M. Sands "The Feynman lectures on physics" Addison-Wesley 1963. Section 51-4. ISBN 0201021153
#[http://dx.doi.org/10.1006/rwos.2001.0129 W.K. Melville "Surface, gravity and capillary waves"], in [http://www.sciencedirect.com/science/referenceworks/9780122274305 "Encyclopedia of Ocean Sciences"], Eds: Steve A. Thorpe and Karl K. Turekian. Elsevier 2001, page 2916. ISBN 978-0-12-227430-5
#[http://dx.doi.org/10.1088/0143-0807/25/1/014 F Behroozi "Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy", European Journal of Physics '''25''' 115 (2004)]
==External links==
*[http://en.wikipedia.org/wiki/Capillary_wave Capillary wave entry in Wikipedia]
*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]
[[Category: Classical thermodynamics ]]
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C60

2012-01-04T05:57:30Z

70.135.118.126:

'''C60''', also known as ''Buckminsterfullerene'' is composed of [[carbon]] atoms.
{{Jmol_general|C60.pdb|C60}}
==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):

:<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>

where

:<math>s=\frac{r}{2a}</math>

:<math>\alpha = \frac{N^2A}{12(2a)^6}</math>

:<math>\beta = \frac{N^2B}{90(2a)^{12}}</math>

where <math>N</math> is the number of atoms on each sphere, i.e. N=60.
==Phase diagram==
<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 C60 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==
<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)]

[[category: models]]
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Buckingham potential

2012-01-04T05:57:19Z

70.135.118.126:

The '''Buckingham potential''' is given by <ref>[http://dx.doi.org/10.1098/rspa.1938.0173 R. A. Buckingham "The Classical Equation of State of Gaseous Helium, Neon and Argon", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences '''168''' pp. 264-283 (1938)]</ref>

:<math>\Phi_{12}(r) = A \exp \left(-Br\right) - \frac{C}{r^6}</math>

where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]], <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math>A</math>, <math>B</math> and <math>C</math> are constants.

The Buckingham potential describes the exchange repulsion, which originates from the Pauli exclusion principle, by a more realistic exponential function of distance, in contrast to the inverse twelfth power used by the [[Lennard-Jones model |Lennard-Jones potential]]. However, since the Buckingham potential remains finite even at very small distances, it runs the risk of an un-physical "Buckingham catastrophe" at short range when used in simulations of charged systems. This occurs when the electrostatic attraction artificially overcomes the repulsive barrier. The Lennard-Jones potential is also about 4 times quicker to compute <ref>[http://dx.doi.org/10.1023/A:1007911511862 David N. J. White "A computationally efficient alternative to the Buckingham potential for molecular mechanics calculations", Journal of Computer-Aided Molecular Design '''11''' pp.517-521 (1997)]</ref> and so is more frequently used in [[Computer simulation techniques | computer simulations]].
==See also==
*[[Exp-6 potential]]
==References==
<references/>
[[category: models]]
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Boltzmann constant

2012-01-04T05:57:07Z

70.135.118.126:

The '''Boltzmann constant''' (<math>k</math> or <math>k_B</math>) is the physical constant relating [[temperature]] to energy.
It is named after the Austrian physicist [[Ludwig Eduard Boltzmann]].
Its experimentally determined value (in SI units, 2010 [http://physics.nist.gov/cgi-bin/cuu/Value?k#mid CODATA] value) is:

:<math>k_B =1.3806488 \times 10^{-23} </math> <math>\left. JK^{-1}\right.</math>

In units with molecular significance it is close to 1, for example see: [[DL_POLY | DL_POLY units]].
==History of Boltzmann's constant==
:''"This constant is often referred to as Boltzmann's constant, although, to my knowledge, Boltzmann himself never introduced it - a peculiar state of affairs, which can be explained by the fact that Boltzmann, as appears from his occasional utterances, never gave thought to the possibility of carrying out an exact measurement of the constant."''
Max Planck, [http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html Nobel Lecture, June 2, 1920]
==Experimental determination of Boltzmann's constant==
Boltzmann's constant can be obtained from the ratio of the [[molar gas constant]] to the [[Avogadro constant]].
The molar gas constant can be obtained via acoustic gas thermometry, and Avogadros constant from either the ''Silicon sphere'', or via the watt balance.
Recently laser spectroscopy has been used to determine the constant (Refs. 3 and 4). Other techniques include Coulomb blockade thermometry (Refs. 5 and 6).
#[http://dx.doi.org/10.1088/0026-1394/22/3/023 L. Storm "Precision Measurements of the Boltzmann Constant",Metrologia '''22''' pp. 229-234 (1986)]
#[http://dx.doi.org/10.1088/0957-0233/17/10/R01 B Fellmuth, Ch Gaiser and J Fischer "Determination of the Boltzmann constant—status and prospects", Measurement Science and Technology '''17''' pp. R145-R159 (2006)]
#[http://dx.doi.org/10.1103/PhysRevLett.98.250801 C. Daussy, M. Guinet, A. Amy-Klein, K. Djerroud, Y. Hermier, S. Briaudeau, Ch. J. Bordé, and C. Chardonnet "Direct Determination of the Boltzmann Constant by an Optical Method", Physical Review Letters '''98''' 250801 (2007)]
#[http://dx.doi.org/10.1103/PhysRevLett.100.200801 G. Casa, A. Castrillo, G. Galzerano, R. Wehr, A. Merlone, D. Di Serafino, P. Laporta, and L. Gianfrani "Primary Gas Thermometry by Means of Laser-Absorption Spectroscopy: Determination of the Boltzmann Constant", Physical Review Letters '''100''' 200801 (2008)]
#[http://dx.doi.org/10.1103/PhysRevLett.73.2903 J. P. Pekola, K. P. Hirvi, J. P. Kauppinen, and M. A. Paalanen "Thermometry by Arrays of Tunnel Junctions", Physical Review Letters '''73''' pp. 2903-2906 (1994)]
#[http://dx.doi.org/10.1103/PhysRevLett.101.206801 Jukka P. Pekola, Tommy Holmqvist, and Matthias Meschke "Primary Tunnel Junction Thermometry", Physical Review Letters '''101''' 206801 (2008)]

==References==
[[Category: Physical constants]]
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Binder cumulant

2012-01-04T05:56:58Z

70.135.118.126:

The '''Binder cumulant''' was introduced by [[Kurt Binder]] in the context of [[Finite size effects |finite size scaling]]. It is a quantity that allows
to locate the critical point and critical exponents. For an [[Ising Models |Ising model]] with zero field, it is given by

:<math>U_4 = 1- \frac{\langle m^4 \rangle }{3\langle m^2 \rangle^2 }</math>

where ''m'' is the [[Order parameters |order parameter]], i.e. the magnetization. It is therefore a fourth order cumulant, related to the kurtosis.
In the [[thermodynamic limit]], where the system size <math>L \rightarrow \infty</math>, <math>U_4 \rightarrow 0</math> for <math>T > T_c</math>, and <math>U_4 \rightarrow 2/3</math> for <math>T < T_c</math>. Thus, the function is discontinuous in this limit. An important observation is that the intersection points of the cumulants for different system sizes usually depend only rather weakly on those sizes, providing a convenient estimate for the value of the [[Critical points |critical temperature]]. Caution is needed in identifying the universality class from
the critical value of the Binder cumulant, because that value depends on boundary condition, system shape, and anisotropy of correlations.

==References==
#[http://dx.doi.org/10.1007/BF01293604 K. Binder "Finite size scaling analysis of ising model block distribution functions", Zeitschrift für Physik B Condensed Matter '''43''' pp. 119-140 (1981)]
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Benjamin Widom

2012-01-04T05:56:48Z

70.135.118.126:

'''Benjamin Widom''' was co-winner of the [[Boltzmann Award]]
in 1998 "for his illuminating studies of the [[statistical mechanics]] of fluids and fluid [[mixtures]] and their [[Interface |interfacial properties]], especially his clear and general formulation of scaling hypotheses for the [[Equations of state |equation of state]] and [[surface tension]] of fluids near [[critical points]]."

Widom was born in Newark, New Jersey. He graduated from New York City's Stuyvesant High School in 1945, and received his B.A. from Columbia University in 1949, followed by his Ph.D from Cornell University in 1953. He became an instructor of chemistry at Cornell in 1954, was appointed assistant professor in 1955 and a full professor in 1963. He was chair of the chemistry department between 1978 and 1981. He was elected a member of the National Academy of Sciences in 1974 and a fellow of the American Academy of Arts and Sciences in 1979.
==Publications list==
Incomplete list:
#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]
#[http://dx.doi.org/10.1080/00268976400100611 H. C. Longuet-Higgins and B. Widom "A rigid sphere model for the melting of argon", Molecular Physics '''8''' pp. 549-556 (1964)]
#[http://dx.doi.org/10.1063/1.1725652 B. Widom "On the Radial Distribution Function in Fluids", Journal of Chemical Physics '''41''' pp. 74-77 (1964)]
#[http://dx.doi.org/10.1126/science.157.3787.375 B. Widom "Intermolecular Forces and the Nature of the Liquid State: Liquids reflect in their bulk properties the attractions and repulsions of their constituent molecules", Science '''157''' pp. 375-382 (1967)]
#[http://dx.doi.org/10.1063/1.1671624 Michael E. Fisher and B. Wiodm "Decay of Correlations in Linear Systems", Journal of Chemical Physics '''50''' 3756 (1969)]
#[http://dx.doi.org/10.1063/1.1673203 B. Widom and J. S. Rowlinson, "New Model for the Study of Liquid–Vapor Phase Transitions", Journal of Chemical Physics '''52''', pp. 1670-1684 (1970)]
#[http://dx.doi.org/10.1021/j100395a005 B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry '''86''' pp. 869 - 872 (1982)]
#B. Widom "Theoretical modeling: An introduction", Berichte der Bunsen-Gesellschaft '''100''' pp. 242- (1996)
#[http://dx.doi.org/10.1021/jp9536460 B. Widom "Theory of Phase Equilibrium", Journal of Physical Chemistry '''100''' pp. 13190–13199 (1996)]
#[http://dx.doi.org/10.1023/B:JOSS.0000033246.14231.e1 Anatoly B. Kolomeisky and B. Widom "A Simplified “Ratchet” Model of Molecular Motors", Journal of Statistical Physics '''93''' pp. 633-645 (1998)]
#[http://dx.doi.org/10.1080/002689799164847 B. Widom "Structure and tension of interfaces", Molecular Physics '''96''' pp. 1019-1026 (1999)]
#[http://dx.doi.org/10.1016/S0378-4371(98)00535-4 B. Widom "What do we know that van der Waals did not know?", Physica A: Statistical Mechanics and its Applications '''263''' pp. 500-515 (1999)]
#[http://dx.doi.org/10.1016/S0378-4371(00)00257-0 H. N. W. Lekkerkerker and B. Widom "An exactly solvable model for depletion phenomena", Physica A '''285''' pp. 483-492 (2000)]
#[http://dx.doi.org/10.1063/1.482049 G. T. Barkema and B. Widom "Model of hydrophobic attraction in two and three dimensions", Journal of Chemical Physics '''113''' pp. 2349-2353 (2000)]
#[http://dx.doi.org/10.1016/S0378-4371(00)00619-1 Volker C. Weiss and B. Widom "Contact angles in sequential wetting: pentane on water", Physica A '''292''' pp. 137-145 (2001)]
#[http://dx.doi.org/10.1039/b304038k B. Widom, P. Bhimalapuram and Kenichiro Koga "The hydrophobic effect", PCCP '''5''' pp. 3085-3093 (2003)]
#[http://dx.doi.org/10.1063/1.1784772 Y. Djikaev and B. Widom "Geometric view of the thermodynamics of adsorption at a line of three-phase contact", Journal of Chemical Physics '''121''' 5602 (2004)]
'''Books'''
*J. S. Rowlinson and B. Widom "Molecular Theory of Capillarity". Dover 2002 (originally: Oxford University Press 1982) ISBN 0486425444
*B. Widom "Statistical Mechanics: A Concise Introduction for Chemists", Cambridge University Press (2002) ISBN 0521009669
==See also==
*[[Fisher-Widom line]]
*[[Widom test-particle method]]
*[[Widom-Rowlinson model]]
==External links==
*[http://www.chem.cornell.edu/faculty/index.asp?fac=45 Homepage of Benjamin Widom]
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Argon

2012-01-04T05:56:29Z

70.135.118.126:

'''Argon''' (Ar) has long been a popular choice for [[Computer simulation techniques |computer simulations]] of simple liquids. Some of the first computer simulations of liquid argon were those of Wood and Parker in 1957 <ref>[http://dx.doi.org/10.1063/1.1743822 W. W. Wood and F. R. Parker "Monte Carlo Equation of State of Molecules Interacting with the Lennard‐Jones Potential. I. A Supercritical Isotherm at about Twice the Critical Temperature", Journal of Chemical Physics '''27''' pp. 720- (1957)]</ref> and of Rahman in 1964 <ref name="Rahman" >[http://dx.doi.org/10.1103/PhysRev.136.A405 A. Rahman "Correlations in the Motion of Atoms in Liquid Argon", Physical Review '''136''' pp. A405–A411 (1964)]</ref>. Sadus and Prausnitz have shown that three-body repulsion makes a significant contribution to [[Idealised models#Three-body potentials|three-body interactions]] in the liquid phase
<ref>[http://dx.doi.org/10.1063/1.471172 Richard J. Sadus and J. M. Prausnitz "Three-body interactions in fluids from molecular simulation: Vapor–liquid phase coexistence of argon", Journal of Chemical Physics '''104''' pp. 4784-4787 (1996)]</ref>
(for use of the [[Axilrod-Teller interaction]] see
<ref>[http://dx.doi.org/10.1103/PhysRevA.45.3659 Phil Attard "Pair-hypernetted-chain closure for three-body potentials: Results for argon with the Axilrod-Teller triple-dipole potential", Physical Review A '''45''' pp. 3659-3669 (1992)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevE.55.2707 J. A. Anta, E. Lomba and M. Lombardero "Influence of three-body forces on the gas-liquid coexistence of simple fluids: The phase equilibrium of argon", Physical Review E '''55''' pp. 2707-2712 (1997)]</ref>).
However, the generic [[Lennard-Jones model]] has been frequently used due to its simplicity; some parameters are quoted in the next section. A specific interatomic potential for Ar has been proposed by Aziz
<ref>[http://dx.doi.org/10.1063/1.466051 Ronald A. Aziz "A highly accurate interatomic potential for argon", Journal of Chemical Physics '''99''' p. 4518 (1993)]</ref>.

==Thermophysical properties (experimental) ==

{| border="1"
|-
| Property <ref>[http://www.webelements.com/argon/physics.html Physical properties of Argon on webelements] </ref> || Temperature || Pressure
|-
| Triple point || 83.8058 K || 69 kPa
|-
| Critical point || 150.87 K || 4.898 MPa
|-
| Melting point || 83.80 K ||
|-
| Boiling point ||87.30 K ||
|}

==Lennard-Jones parameters==
[[Image:Lennard-Jones.png|thumb| The Lennard-Jones model for argon (Rowley, Nicholson and Parsonage parameters).]]
A selection of parameters for the [[Lennard-Jones model]] for liquid argon are listed in the following table:
{| border="1"
|-
| Authors || <math>\epsilon/k_B</math> (K) || <math>\sigma</math> (nm)|| Reference (year)
|-
| Rahman || 120 || 0.34 || <ref name="Rahman"> </ref> (1964)
|-
|Barker, Fisher and Watts ||142.095 || 0.33605 || <ref>[http://dx.doi.org/10.1080/00268977100101821 J. A. Barker, R. A. Fisher and R. O. Watts "Liquid argon: Monte carlo and molecular dynamics calculations", Molecular Physics '''21''' pp. 657-673 (1971)]</ref> (1971)
|-
| Rowley, Nicholson and Parsonage || 119.8 || 0.3405 || <ref>[http://dx.doi.org/10.1016/0021-9991(75)90042-X L. A. Rowley, D. Nicholson and N. G. Parsonage "Monte Carlo grand canonical ensemble calculation in a gas-liquid transition region for 12-6 Argon", Journal of Computational Physics '''17''' pp. 401-414 (1975)]</ref> (1975)
|-
| White || 125.7 || 0.3345 || <ref>[http://dx.doi.org/10.1063/1.479848 John A. White "Lennard-Jones as a model for argon and test of extended renormalization group calculations", Journal of Chemical Physics '''111''' pp. 9352-9356 (1999)]</ref> parameter set #4 (1999)
|}

==Buckingham potential==
The [[Buckingham potential]] for argon is given by (Eq. 27 <ref>[http://dx.doi.org/10.1098/rspa.1938.0173 R. A. Buckingham "The Classical Equation of State of Gaseous Helium, Neon and Argon", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences '''168''' pp. 264-283 (1938)]</ref>):
:<math>E(r) = 1.69 \times 10^{-8} e^{-r/0.273} -102 \times 10^{-12} r^{-6} </math>
where <math>E</math> is in ergs ( 10−7 J) and <math>r</math> in Å.
==BBMS potential==
The Bobetic-Barker-Maitland-Smith potential <ref>[http://dx.doi.org/10.1103/PhysRevB.2.4169 M. V. Bobetic and J. A. Barker "Lattice Dynamics with Three-Body Forces: Argon", Physical Review B '''2''' 4169-4175 (1970)]</ref> <ref>[http://dx.doi.org/10.1080/00268977100103181 G. C. Maitland and E. B. Smith "The intermolecular pair potential of argon", Molecular Physics '''22''' pp. 861-868 (1971)]</ref>.
==Dymond-Alder pair potential==
<ref>[http://dx.doi.org/10.1063/1.1671724 J. H. Dymond and B. J. Alder "Pair Potential for Argon", Journal of Chemical Physics '''51''' pp. 309-320 (1969)]</ref>
==Radial distribution function==
[[Radial distribution function]] <ref>[http://dx.doi.org/10.1103/PhysRevA.7.2130 J. L. Yarnell, M. J. Katz, R. G. Wenzel and S. H. Koenig "Structure Factor and Radial Distribution Function for Liquid Argon at 85°K", Physical Review A '''7''' pp. 2130-2144 (1973)]</ref>
==Nucleation==
<ref>[http://dx.doi.org/10.1063/1.3474945 Matthew J. McGrath, Julius N. Ghogomu, Narcisse T. Tsona, J. Ilja Siepmann, Bin Chen, Ismo Napari1, and Hanna Vehkamäki "Vapor-liquid nucleation of argon: Exploration of various intermolecular potentials", Journal of Chemical Physics '''133''' 084106 (2010)]</ref>
==Quantum simulations==
<ref>[http://dx.doi.org/10.1080/00268978900100811 J. R. Melrose and K. Singer "An investigation of supercooled Lennard-Jones argon by quantum mechanical and classical Monte Carlo simulation", Molecular Physics '''66''' 1203-1214 (1989)]</ref>
==Virial equation of state==
[[Virial equation of state]] <ref>[http://dx.doi.org/10.1063/1.3627151 Benjamin Jäger, Robert Hellmann, Eckard Bich, and Eckhard Vogel "Ab initio virial equation of state for argon using a new nonadditive three-body potential", Journal of Chemical Physics '''135''' 084308 (2011)]</ref>
==References==
<references/>
'''Related material'''
*[http://dx.doi.org/10.1080/00268976400100611 H. C. Longuet-Higgins and B. Widom "A rigid sphere model for the melting of argon", Molecular Physics '''8''' pp. 549-556 (1964)]
*[http://dx.doi.org/10.1080/00268976800100721 D. Henderson and J. A. Barker "On the solidification of argon", Molecular Physics '''14''' pp. 587-589 (1968)]
*[http://dx.doi.org/10.1103/PhysRevA.5.2238 F. Lado "Numerical Calculation of the Density Autocorrelation Function for Liquid Argon", Physical Review A '''5''' pp. 2238-2244 (1972)]
*[http://dx.doi.org/10.1016/j.molliq.2009.09.009 Ali Asghar Davoodi and Farzaneh Feyzi "A new approach for long range corrections in molecular dynamics simulation with application to calculation of argon properties", Journal of Molecular Liquids '''150''' pp. 33-38 (2009)]

[[category: models]]
{{Numeric}}
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AMBER forcefield

2012-01-04T05:53:17Z

70.135.118.126:

:''This page contains information about the AMBER [[Force fields |forcefield]]. See [[AMBER_--_Assisted_Model_Building_with_Energy_Refinement | the AMBER package]] for information about the computer code.''
{{Stub-general}}
{{Numeric}}
==Force field==
<math>E_{\rm total} = \sum_{\rm bonds} K_r (r - r_{eq})^2
+ \sum_{\rm angles} K_\theta (\theta - \theta_{eq})^2
+ \sum_{\rm dihedrals} {V_n \over 2}
[1 + {\rm cos}(n\phi - \gamma)]

+ \sum_{i<j} \left [ {A_{ij} \over R_{ij}^{12}} -
{B_{ij} \over R_{ij}^6} +
{q_iq_j \over \epsilon R_{ij}}
\right ]
+ \sum_{\rm H-bonds}
\left [ {C_{ij} \over R_{ij}^{12}} -
{D_{ij} \over R_{ij}^{10}} \right ] </math>
==Parameters==
====ff94====
*[http://dx.doi.org/10.1021/ja00124a002 Wendy D. Cornell, Piotr Cieplak, Christopher I. Bayly, Ian R. Gould, Kenneth M. Merz, David M. Ferguson, David C. Spellmeyer, Thomas Fox, James W. Caldwell, and Peter A. Kollman "A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules", Journal of the American Chemical Society '''117''' pp 5179 - 5197 (1995)]
====ff96====
====ff98====
====ff99====
Param99 was developed for organic and biological molecules using the restrained electrostatic potential (RESP) approach to derive the partial charges:
*[http://www3.interscience.wiley.com/cgi-bin/abstract/72511509/ABSTRACT Junmei Wang, Piotr Cieplak, Peter A. Kollman "How well does a restrained electrostatic potential (RESP) model perform in calculating conformational energies of organic and biological molecules?", Journal of Computational Chemistry '''21''' pp. 1049-1074 (2000)]

====ff02====
The ff02 force field is a polarisable variant of ff99.
The charges were determined at the B3LYP/cc-
pVTZ//HF/6-31g* level, and hence are more like "gas-phase"
charges.
*[http://dx.doi.org/10.1002/jcc.1065 Piotr Cieplak, James Caldwell and Peter Kollman "Molecular mechanical models for organic and biological systems going beyond the atom centered two body additive approximation: aqueous solution free energies of methanol and N-methyl acetamide, nucleic acid base, and amide hydrogen bonding and chloroform/water partition coefficients of the nucleic acid bases", Journal of Computational Chemistry '''22''' pp. 1048-1057 (2001)]
====ff02EP====
==References==
#[http://dx.doi.org/10.1016/0010-4655(95)00041-D David A. Pearlman, David A. Case, James W. Caldwell, Wilson S. Ross, Thomas E. Cheatham, Steve DeBolt, David Ferguson, George Seibel and Peter Kollman "AMBER, a package of computer programs for applying molecular mechanics, normal mode analysis, molecular dynamics and free energy calculations to simulate the structural and energetic properties of molecules", Computer Physics Communications '''91''' pp. 1-41 (1995)]
==External links==
*[http://amber.scripps.edu/dbase.html AMBER parameters]
[[category:Force fields]]
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