Radial distribution function: Difference between revisions

Revision as of 22:40, 14 November 2018

Revision as of 17:13, 20 November 2018

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The radial distribution function is an important measure because several key thermodynamic properties, such as potential energy and pressure can be calculated from it.

For a 3-D system where particles interact via pairwise potentials, the potential energy of the system can be calculated as follows:<ref name=softmatter>{{cite book|last=Frenkel|first=Daan; Smit, Berend|title=Understanding molecular simulation from algorithms to applications|year=2002|publisher=Academic Press|___location=San Diego|isbn=0122673514978-0122673511|edition=2nd}}</ref>

<math>PE=\frac{N}{2}4\pi\rho\int^{\infty}_0r^2u(r)g(r)dr </math>

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One can determine <math>g(r)</math> indirectly (via its relation with the structure factor <math>S(q)</math>) using [[neutron scattering]] or [[x-ray scattering]] data. The technique can be used at very short length scales (down to the atomic level<ref>{{Cite journal | last1 = Yarnell | first1 = J. | last2 = Katz | first2 = M. | last3 = Wenzel | first3 = R. | last4 = Koenig | first4 = S. | title = Structure Factor and Radial Distribution Function for Liquid Argon at 85 °K | doi = 10.1103/PhysRevA.7.2130 | journal = Physical Review A | volume = 7 | issue = 6 | pages = 2130 | year = 1973 | pmid = | pmc = |bibcode = 1973PhRvA...7.2130Y }}</ref>) but involves significant space and time averaging (over the sample size and the acquisition time, respectively). In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals<ref>{{Cite journal | last1 = Gingrich | first1 = N. S. | last2 = Heaton | first2 = L. | doi = 10.1063/1.1731688 | title = Structure of Alkali Metals in the Liquid State | journal = The Journal of Chemical Physics | volume = 34 | issue = 3 | pages = 873 | year = 1961 | pmid = | pmc = |bibcode = 1961JChPh..34..873G }}</ref> to charged colloids.<ref>{{Cite journal | last1 = Sirota | first1 = E. | last2 = Ou-Yang | first2 = H. | last3 = Sinha | first3 = S. | last4 = Chaikin | first4 = P. | last5 = Axe | first5 = J. | last6 = Fujii | first6 = Y. | doi = 10.1103/PhysRevLett.62.1524 | title = Complete phase diagram of a charged colloidal system: A synchro- tron x-ray scattering study | journal = Physical Review Letters | volume = 62 | issue = 13 | pages = 1524–1527 | year = 1989 | pmid = 10039696| pmc = |bibcode = 1989PhRvL..62.1524S }}</ref> It should be noted that going from the experimental <math>S(q)</math> to <math>g(r)</math> is not straightforward and the analysis can be quite involved.<ref>{{Cite journal | last1 = Pedersen | first1 = J. S. | doi = 10.1016/S0001-8686(97)00312-6 | title = Analysis of small-angle scattering data from colloids and polymer solutions: Modeling and least-squares fitting | journal = Advances in Colloid and Interface Science | volume = 70 | pages = 171–201 | year = 1997 | pmid = | pmc = }}</ref>

It is also possible to calculate <math>g(r)</math> directly by extracting particle positions from traditional or confocal microscopy.<ref>{{Cite journal | last1 = Crocker | first1 = J. C.| last2 = Grier | first2 = D. G.| doi = 10.1006/jcis.1996.0217 | title = Methods of Digital Video Microscopy for Colloidal Studies | journal = Journal of Colloid and Interface Science | volume = 179 | issue = 1| pages = 298–310| year = 1996 | pmid = | pmc = | bibcode = 1996JCIS..179..298C}}</ref> This technique is limited to particles large enough for optical detection (in the micrometer range), but it has the advantage of being time-resolved so that, aside from the statical information, it also gives access to dynamical parameters (e.g. [[diffusion constant]]s<ref>{{Cite journal | last1 = Nakroshis | first1 = P. | last2 = Amoroso | first2 = M. | last3 = Legere | first3 = J. | last4 = Smith | first4 = C. | title = Measuring Boltzmann's constant using video microscopy of Brownian motion | doi = 10.1119/1.1542619 | journal = American Journal of Physics | volume = 71 | issue = 6 | pages = 568 | year = 2003 | pmid = | pmc = |bibcode = 2003AmJPh..71..568N }}</ref>) and also space-resolved (to the level of the individual particle), allowing it to reveal the morphology and dynamics of local structures in colloidal crystals,<ref>{{Cite journal | last1 = Gasser | first1 = U. | last2 = Weeks | first2 = E. R. | last3 = Schofield | first3 = A. | last4 = Pusey | first4 = P. N. | last5 = Weitz | first5 = D. A. | title = Real-Space Imaging of Nucleation and Growth in Colloidal Crystallization | doi = 10.1126/science.1058457 | journal = Science | volume = 292 | issue = 5515 | pages = 258–262 | year = 2001 | pmid = 11303095| pmc = |bibcode = 2001Sci...292..258G }}</ref> glasses,<ref>{{Cite journal | last1 = Weeks | first1 = E. R. | last2 = Crocker | first2 = J. C. | last3 = Levitt | first3 = A. C. | last4 = Schofield | first4 = A. | last5 = Weitz | first5 = D. A. | title = Three-Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition | doi = 10.1126/science.287.5453.627 | journal = Science | volume = 287 | issue = 5453 | pages = 627–631 | year = 2000 | pmid = 10649991| pmc = |bibcode = 2000Sci...287..627W }}</ref> gels,<ref>{{Cite journal | last1 = Cipelletti | first1 = L. | last2 = Manley | first2 = S. | last3 = Ball | first3 = R. C. | last4 = Weitz | first4 = D. A. | title = Universal Aging Features in the Restructuring of Fractal Colloidal Gels | doi = 10.1103/PhysRevLett.84.2275 | journal = Physical Review Letters | volume = 84 | issue = 10 | pages = 2275–2278 | year = 2000 | pmid = 11017262| pmc = |bibcode = 2000PhRvL..84.2275C }}</ref><ref>{{Cite journal | last1 = Varadan | first1 = P. | last2 = Solomon | first2 = M. J. | doi = 10.1021/la026303j | title = Direct Visualization of Long-Range Heterogeneous Structure in Dense Colloidal Gels | journal = Langmuir | volume = 19 | issue = 3 | pages = 509 | year = 2003 | pmid = | pmc = }}</ref> and hydrodynamic interactions.<ref>{{Cite journal | last1 = Gao | first1 = C. | last2 = Kulkarni | first2 = S. D. | last3 = Morris | first3 = J. F. | last4 = Gilchrist | first4 = J. F. | title = Direct investigation of anisotropic suspension structure in pressure-driven flow | doi = 10.1103/PhysRevE.81.041403 | journal = Physical Review E | volume = 81 | issue = 4 | pages = 041403 | year = 2010 | pmid = 20481723| pmc = |bibcode = 2010PhRvE..81d1403G }}</ref>

Direct visualization of a full (distance-dependent and angle-dependent) pair correlation function was achieved by a [[scanning tunneling microscope|scanning tunneling microscopy]] in the case of 2D molecular gases <ref>{{Cite journal|last=Matvija|first=Peter|last2=Rozbořil|first2=Filip|last3=Sobotík|first3=Pavel|last4=Ošťádal|first4=Ivan|last5=Kocán|first5=Pavel|date=|title=Pair correlation function of a 2D molecular gas directly visualized by scanning tunneling microscopy|url=http://pubs.acs.org/doi/abs/10.1021/acs.jpclett.7b01965|journal=The Journal of Physical Chemistry Letters|volume=8|issue=17|pages=4268–4272|doi=10.1021/acs.jpclett.7b01965|viapmid=28830146|year=2017}}</ref>.

==Higher-order correlation functions==

Higher-order distribution functions <math>\textstyle g^{(k)}</math> with <math>\textstyle k > 2</math> were less studied, since they are generally less important for the thermodynamics of the system; at the same time, they are not accessible by conventional scattering techniques. They can however be measured by [[coherent scattering|coherent X-ray scattering]] and are interesting insofar as they can reveal local symmetries in disordered systems.<ref>{{Cite journal | last1 = Wochner | first1 = P. | last2 = Gutt | first2 = C. | last3 = Autenrieth | first3 = T. | last4 = Demmer | first4 = T. | last5 = Bugaev | first5 = V. | last6 = Ortiz | first6 = A. D. | last7 = Duri | first7 = A. | last8 = Zontone | first8 = F. | last9 = Grubel | first9 = G. | doi = 10.1073/pnas.0905337106 | last10 = Dosch | first10 = H. | title = X-ray cross correlation analysis uncovers hidden local symmetries in disordered matter | journal = Proceedings of the National Academy of Sciences | volume = 106 | issue = 28 | pages = 1151111511–4 | year = 2009 | pmid = 20716512| pmc = 2703671|bibcode = 2009PNAS..10611511W }}</ref>

==References==