View source for Fiveling

{{Short description|Five crystals arranged round a common axis}}
{{good article}}
{{Use dmy dates|date=May 2024}}
{{Use American English|date=May 2024}}
[[File:PtFe1.2_nanoparticle.jpg|thumb|Decahedral PtFe1.2 nanoparticle.<ref>{{Cite journal |last1=Jang |first1=Ji-Hoon |last2=Lee |first2=Eunjik |last3=Park |first3=Jinwoo |last4=Kim |first4=Gunn |last5=Hong |first5=Suklyun |last6=Kwon |first6=Young-Uk |date=2013 |title=Rational syntheses of core-shell Fex@Pt nanoparticles for the study of electrocatalytic oxygen reduction reaction |journal=Scientific Reports |language=en |volume=3 |issue=1 |page=2872 |doi=10.1038/srep02872 |issn=2045-2322 |pmc=3791448 |pmid=24096587}}</ref>]]
A '''fiveling''', also known as a '''decahedral nanoparticle''', a '''multiply-twinned particle''' (MTP), a '''pentagonal nanoparticle''', a '''pentatwin''', or a '''five-fold twin''' is a type of [[twinned crystal]] that can exist at sizes ranging from nanometers to [[millimetre]]s. It contains five different [[single crystal]]s arranged around a common axis. In most cases each unit has a [[Cubic crystal system#Bravais lattice|face centered cubic]] (fcc) arrangement of the atoms, although they are also known for other types of [[crystal structure]].

They [[nucleate]] at quite small sizes in the [[nanometer]] range, but can be grown much larger. They have been found in [[mineral]] [[crystal]]s{{efn|name=crystal}} excavated from mines such as [[pentagonite]]<ref name=":6" /> or native gold from Ukraine,<ref>{{Cite journal |last1=Kvasnifsa |first1=V. N. |last2=Kuznetsov |first2=Yu. A. |last3=Latysh |first3=I. K. |date=1981 |title=Crystal morphology of native gold from some ore regions of the Ukraine |url=https://www.tandfonline.com/doi/full/10.1080/00206818209467235 |journal=International Geology Review |language=en |volume=23 |issue=2 |pages=227–232 Figure 5 |doi=10.1080/00206818209467235 |bibcode=1981IGRv...23..227K |issn=0020-6814}}</ref> in rods of metals grown via [[electrochemical]] processes and in nanoparticles produced by the condensation of metals either onto substrates or in inert gases. They have been investigated for their potential uses in areas such as improving the efficiency of [[solar cell]] or [[heterogeneous catalysis]] for more efficient production of chemicals. Information about them is distributed across a diverse range of scientific disciplines, mainly [[chemistry]], [[materials science]], [[mineralogy]], [[nanomaterials]] and [[physics]]. Because many different names have been used, sometimes the information in the different disciplines or within any one discipline is fragmented and overlapping.

At small sizes in the nanometer range, up to millimetres in size, with fcc metals they often have a combination of {111} and {100} facets, a low energy shape called a '''Marks decahedron'''.<ref name=":7" /><ref>{{Cite web |last=Doye |first=Jonathan |date=1996 |title=The Structure, Thermodynamics and Dynamics of Atomic Clusters |url=http://doye.chem.ox.ac.uk/jon/PhD2/node8.html |access-date=2024-05-09 |website=doye.chem.ox.ac.uk}}</ref> Relative to a [[single crystal]], at small sizes a fiveling can be a lower energy structure due to having more low energy [[Face (geometry)|surface facets]].{{efn|name=facets}} Balancing this there is an energy cost due to [[Elastic strain energy|elastic strains]] to close an angular gap ([[disclination]]), which makes them higher in energy at larger sizes. They can be the most stable structure in some intermediate sizes, but they can be one among many in a population of different structures due to a combination of coexisting nanoparticles and kinetic growth factors. The temperature, gas environment and [[chemisorption]] can play an important role in both their [[thermodynamic stability]] and growth. While they are often symmetric, they can also be asymmetric with the disclination not in the center of the particle.

== History ==
[[File:Rose-Marks_Decahedron.gif|thumb|Redrawn version of 1831 sketch of a gold fiveling by Rose,<ref name=":11" /> which is a Marks Decahedron<ref name=":4" /><ref name=":12" /> with <math>\gamma_{111} \approx 0.7\gamma_{100}</math>.]]
Dating back to the nineteenth century there are reports of these particles by authors such as [[Jacques-Louis Bournon]] in 1813 for [[marcasite]],<ref name=":25">{{Cite book |last=Comte de Bournon |first=Jacques-Louis |url=https://gallica.bnf.fr/ark:/12148/bpt6k9798806q |title=Catalogue de la collection minéralogique du comte de Bournon,... faites par lui-même . Et dans lequel sont placés plusieurs observations et faits intéressants... ainsi qu'une réponse au mémoire de M. l'abbé Haüy concernant la simplicité des lois auxquelles est soumise la structure des cristaux, etc. |date=1813 |publisher=L. Deconchy |pages=301–308 |language=EN}}</ref><ref>{{Cite book |last=Comte de Bournon |first=Jacques-Louis |url=https://gallica.bnf.fr/ark:/12148/bpt6k9807493z |title=Catalogue de la collection minéralogique du comte de Bournon,... faites par lui-même . Et dans lequel sont placés plusieurs observations et faits intéressants... ainsi qu'une réponse au mémoire de M. l'abbé Haüy concernant la simplicité des lois auxquelles est soumise la structure des cristaux, etc. |date=1813 |publisher=L. Deconchy |pages=plates VIII and esp. IX, fig 164–168 |language=EN}}</ref> and [[Gustav Rose]] in 1831 for gold.<ref name=":11">{{Cite journal |last=Rose |first=Gustav |date=1831 |title=Ueber die Krystallformen des Goldes und des Silbers |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.18310991003 |journal=Annalen der Physik |language=en |volume=99 |issue=10 |pages=196–204 |doi=10.1002/andp.18310991003 |bibcode=1831AnP....99..196R |issn=0003-3804}}</ref>  In mineralogy and the [[crystal twinning]] literature they are referred to as a type of cyclic twin where a number of identical single crystal units are arranged in a ring-like pattern where they all join at a common point or line.<ref>{{Cite web |last=Perkins |first=Dexter |date=2022 |title=4.4.6: Crystal Twinning |url=https://geo.libretexts.org/Bookshelves/Geology/Mineralogy_(Perkins_et_al.)/04%3A_Crystals_and_Crystallization/4.04%3A_Factors_Controlling_Crystal_Size_and_Perfection/4.4.06%3A_Crystal_Twinning |access-date=2024-03-27 |website=Geosciences LibreTexts |language=en}}</ref> The name '''''{{linktext|fiveling}}''''' comes from them having five members (single crystals).<ref>{{Cite web |title=Definition of FIVELING |url=https://www.merriam-webster.com/dictionary/fiveling |access-date=2024-03-27 |website=www.merriam-webster.com |language=en}}</ref>  Fivelings have also been described as a type of [[macle]] twinning.<ref>{{cite book |last1=Collins |first1=Joseph Henry |url=https://books.google.com/books?id=fg8OlL8A5a0C |title=Mineralogy. With ... Illustrations, Volym 1 |date=1878 |publisher=[[National Library of Scotland]] |page=120}}</ref> The older literature was mainly observational, with information on many materials documented by [[Victor Mordechai Goldschmidt]] in his ''Atlas der Kristallformen''.<ref>{{Cite book |last=Goldschmidt |first=Victor |url=https://catalog.hathitrust.org/Record/009540159 |title=Atlas der Krystallformen |date=1913–1923 |publisher=C. Winters |location=Heidelberg |trans-title=Atlas of Crystal Forms}}</ref> Drawings are available showing their presence in marcasite, gold, silver, copper and diamond. New mineral forms with a fiveling structure continue to be found, for instance [[pentagonite]], whose structure was first decoded in 1973, is named because it is often found with the five-fold twinning.<ref name=":6">{{Cite journal |last1=Staples |first1=L. W. |last2=Evans |first2=H. T. |last3=Lindsay |first3=J. R. |date=1973 |title=Cavansite and Pentagonite, New Dimorphous Calcium Vanadium Silicate Minerals from Oregon |journal=American Mineralogist |volume=58 |issue=5–6 |pages=405–411}}</ref><ref>{{Cite journal |last=White |first=John |date=2002 |title=Let's Get It Right: Cavansite or Pentagonite? |url=http://dx.doi.org/10.1080/00357529.2002.9925646 |journal=Rocks & Minerals |volume=77 |issue=4 |pages=274–275 |doi=10.1080/00357529.2002.9925646 |bibcode=2002RoMin..77..274W |issn=0035-7529}}</ref>

Most modern analysis started with the observation of these particles by Shozo Ino and Shiro Ogawa in 1966-67,<ref name=":18">{{Cite journal |last=Ino |first=Shozo |date=1966 |title=Epitaxial Growth of Metals on Rocksalt Faces Cleaved in Vacuum. II. Orientation and Structure of Gold Particles Formed in Ultrahigh Vacuum |url=https://journals.jps.jp/doi/10.1143/JPSJ.21.346 |journal=Journal of the Physical Society of Japan |language=en |volume=21 |issue=2 |pages=346–362 |doi=10.1143/JPSJ.21.346 |bibcode=1966JPSJ...21..346I |issn=0031-9015}}</ref><ref name=":19">{{Cite journal |last1=Ino |first1=Shozo |last2=Ogawa |first2=Shiro |date=1967 |title=Multiply Twinned Particles at Earlier Stages of Gold Film Formation on Alkalihalide Crystals |url=https://journals.jps.jp/doi/10.1143/JPSJ.22.1365 |journal=Journal of the Physical Society of Japan |language=en |volume=22 |issue=6 |pages=1365–1374 |doi=10.1143/JPSJ.22.1365 |bibcode=1967JPSJ...22.1365I |issn=0031-9015}}</ref> and independently but slightly later (which they acknowledged) in work by John Allpress and John Veysey Sanders.<ref name=":0">{{Cite journal |last1=Allpress |first1=J.G. |last2=Sanders |first2=J.V. |date=1967 |title=The structure and orientation of crystals in deposits of metals on mica |url=https://linkinghub.elsevier.com/retrieve/pii/0039602867900623 |journal=Surface Science |language=en |volume=7 |issue=1 |pages=1–25 |doi=10.1016/0039-6028(67)90062-3|bibcode=1967SurSc...7....1A }}</ref> In both cases these were for [[vacuum deposition]] of metal onto substrates in very clean ([[ultra-high vacuum]]) conditions, where nanoparticle islands of size 10-50&nbsp;nm were formed during [[thin film]] growth. Using [[Transmission electron microscopy|transmission electron microscopy and diffraction]] these authors demonstrated the presence of the five single crystal units in the particles, and also the twin relationships. They also observed single crystals and a related type of [[Icosahedral twins|icosahedral nanoparticle]]. They called the five-fold and icosahedral crystals '''multiply twinned particles''' ('''MTPs'''). In the early work near perfect decahedron (pentagonal bipyramid) and icosahedron shapes were formed, so they were called '''decahedral MTPs''' or '''icosahedral MTPs''', the names connecting to the decahedral (<math>D_{5h}</math>) and [[Icosahedral symmetry|icosahedral]] (<math>I_h</math>) [[Point groups in three dimensions|point group symmetries]].{{efn|name=symmetry}} Parallel, and apparently independent there was work on larger metal [[Whisker (metallurgy)|whiskers]] ([[nanowire]]s) which sometimes showed a very similar five-fold structure,<ref>{{Cite journal |last=Schwoebel |first=Richard L. |date=1966 |title=Anomalous Growth of Gold from the Vapor Phase |url=https://pubs.aip.org/jap/article/37/6/2515/166113/Anomalous-Growth-of-Gold-from-the-Vapor-Phase |journal=Journal of Applied Physics |language=en |volume=37 |issue=6 |pages=2515–2516 |doi=10.1063/1.1708849 |bibcode=1966JAP....37.2515S |issn=0021-8979}}</ref><ref>{{Cite journal |last1=Smit |first1=J. |last2=Ogburn |first2=F. |last3=Bechtoldt |first3=C. J. |date=1968 |title=Multiple Twin Structures in Electrodeposited Silver Dendrites |url=https://iopscience.iop.org/article/10.1149/1.2411207 |journal=Journal of the Electrochemical Society |language=en |volume=115 |issue=4 |pages=371 |doi=10.1149/1.2411207|bibcode=1968JElS..115..371S }}</ref> an occurrence reported in 1877 by [[Gerhard vom Rath]].<ref>{{Cite journal |last=Rath |first=G. vom |date=1877 |title=Mineralogische Mittheilungen |url=https://www.degruyter.com/document/doi/10.1524/zkri.1877.1.1.1/html |journal=Zeitschrift für Kristallographie - Crystalline Materials |language=en |volume=1 |issue=1–6 |pages=1–17 |doi=10.1524/zkri.1877.1.1.1 |issn=2196-7105}}</ref> There was fairly extensive analysis following this, particularly for the nanoparticles, both of their internal structure by some of the first electron microscopes that could image at the atomic scale,<ref name=":1">{{Cite journal |last=Komoda |first=Tsutomu |date=1968 |title=Study on the Structure of Evaporated Gold Particles by Means of a High Resolution Electron Microscope |url=https://iopscience.iop.org/article/10.1143/JJAP.7.27 |journal=Japanese Journal of Applied Physics |volume=7 |issue=1 |pages=27 |doi=10.1143/JJAP.7.27 |bibcode=1968JaJAP...7...27K |issn=0021-4922}}</ref> and by various continuum or atomic models as cited later.

Following this early work there was a large effort, mainly in Japan, to understand what were then called "fine particles", but would now be called nanoparticles. By heating up different elements so atoms evaporated and were then condensed in an inert argon atmosphere, fine particles of almost all the elemental solids were made and then analyzed using electron microscopes. The decahedral particles were found for all face centered cubic materials and a few others, often together with other shapes.<ref name=":17">{{Cite journal |last1=Hayashi |first1=Takayoshi |last2=Ohno |first2=Takehisa |last3=Yatsuya |first3=Shigeki |last4=Uyeda |first4=Ryozi |date=1977 |title=Formation of Ultrafine Metal Particles by Gas-Evaporation Technique. IV. Crystal Habits of Iron and Fcc Metals, Al, Co, Ni, Cu, Pd, Ag, In, Au and Pb |url=https://iopscience.iop.org/article/10.1143/JJAP.16.705 |journal=Japanese Journal of Applied Physics |language=en |volume=16 |issue=5 |pages=705–717 |doi=10.1143/JJAP.16.705 |bibcode=1977JaJAP..16..705H |issn=0021-4922}}</ref><ref name=":30">{{Cite journal |last=Iijima |first=Sumio |date=1987 |title=Fine Particles of Silicon. II. Decahedral Multiply-Twinned Particles |url=https://iopscience.iop.org/article/10.1143/JJAP.26.365 |journal=Japanese Journal of Applied Physics |volume=26 |issue=3R |pages=365 |doi=10.1143/JJAP.26.365 |bibcode=1987JaJAP..26..365I |issn=0021-4922}}</ref><ref name=":26">{{Cite journal |last1=Zhou |first1=Shan |last2=Zhao |first2=Ming |last3=Yang |first3=Tung-Han |last4=Xia |first4=Younan |date=2019 |title=Decahedral nanocrystals of noble metals: Synthesis, characterization, and applications |journal=Materials Today |volume=22 |pages=108–131 |doi=10.1016/j.mattod.2018.04.003 |issn=1369-7021|doi-access=free }}</ref>{{anchor|LJ75}}
[[File:Side view of LJ cluster with 75 atoms.gif|thumb|Calculated minimum energy decahedral structure for 75 atoms with a [[Lennard-Jones potential]], an atomistic version of a Marks decahedron.<ref name=":51" />]]While there was some continuing work over the following decades, it was with the [[National Nanotechnology Initiative]]<ref>{{Cite web |title=Nanotechnology Timeline |url=https://www.nano.gov/timeline |access-date=2020-12-05 |website=nano.gov}}</ref> that substantial interest was reignited. At the same time terms such as '''pentagonal nanoparticle''', '''pentatwin''', or '''five-fold twin''' became common in the literature, together with the earlier names. A large number of different methods have now been published for fabricating fivelings, sometimes with a high yield but often as part of a larger population of different shapes.<ref name=":38">{{Cite journal |last1=Marks |first1=L D |last2=Peng |first2=L |date=2016 |title=Nanoparticle shape, thermodynamics and kinetics |url=https://iopscience.iop.org/article/10.1088/0953-8984/28/5/053001 |journal=Journal of Physics: Condensed Matter |volume=28 |issue=5 |pages=053001 |doi=10.1088/0953-8984/28/5/053001 |pmid=26792459 |bibcode=2016JPCM...28e3001M |issn=0953-8984}}</ref> These range from colloidal solution methods<ref>{{Cite journal |last1=Jin |first1=Rongchao |last2=Zeng |first2=Chenjie |last3=Zhou |first3=Meng |last4=Chen |first4=Yuxiang |date=2016 |title=Atomically Precise Colloidal Metal Nanoclusters and Nanoparticles: Fundamentals and Opportunities |url=https://pubs.acs.org/doi/10.1021/acs.chemrev.5b00703 |journal=Chemical Reviews |language=en |volume=116 |issue=18 |pages=10346–10413 |doi=10.1021/acs.chemrev.5b00703 |pmid=27585252 |issn=0009-2665}}</ref> to different deposition approaches.<ref name=":17" /><ref>{{Cite journal |last1=Elechiguerra |first1=Jose Luis |last2=Reyes-Gasga |first2=Jose |last3=Yacaman |first3=Miguel Jose |date=2006 |title=The role of twinning in shape evolution of anisotropic noble metal nanostructures |url=http://xlink.rsc.org/?DOI=b607128g |journal=Journal of Materials Chemistry |language=en |volume=16 |issue=40 |pages=3906 |doi=10.1039/b607128g |issn=0959-9428}}</ref> It is documented that fivelings occur frequently for diamond,<ref>{{Cite journal |last1=Matsumoto |first1=Seiichiro |last2=Matsui |first2=Yoshio |date=1983 |title=Electron microscopic observation of diamond particles grown from the vapour phase |url=http://link.springer.com/10.1007/BF00542075 |journal=Journal of Materials Science |language=en |volume=18 |issue=6 |pages=1785–1793 |doi=10.1007/BF00542075 |bibcode=1983JMatS..18.1785M |issn=0022-2461}}</ref><ref>{{Cite journal |last1=Bühler |first1=Jürgen |last2=Prior |first2=Yehiam |date=2000 |title=Study of morphological behavior of single diamond crystals |url=https://linkinghub.elsevier.com/retrieve/pii/S0022024899006582 |journal=Journal of Crystal Growth |language=en |volume=209 |issue=4 |pages=779–788 |doi=10.1016/S0022-0248(99)00658-2|bibcode=2000JCrGr.209..779B }}</ref> gold and silver,<ref name=":27">{{Cite journal |last1=Rogers |first1=Blake |last2=Lehr |first2=Alexander |last3=Velázquez-Salazar |first3=J. Jesús |last4=Whetten |first4=Robert |last5=Mendoza-Cruz |first5=Ruben |last6=Bazan-Diaz |first6=Lourdes |last7=Bahena-Uribe |first7=Daniel |last8=José Yacaman |first8=Miguel |date=2023 |title=Decahedra and Icosahedra Everywhere: The Anomalous Crystallization of Au and Other Metals at the Nanoscale |url=https://onlinelibrary.wiley.com/doi/10.1002/crat.202200259 |journal=Crystal Research and Technology |language=en |volume=58 |issue=4 |doi=10.1002/crat.202200259 |bibcode=2023CryRT..5800259R |osti=2417752 |issn=0232-1300}}</ref> sometimes for copper<ref>{{Cite journal |last1=Ogburn |first1=F. |last2=Paretzkin |first2=B. |last3=Peiser |first3=H. S. |date=1964 |title=Pseudopentagonal twins in electrodeposited copper dendrites |url=https://scripts.iucr.org/cgi-bin/paper?S0365110X64002006 |journal=Acta Crystallographica |volume=17 |issue=6 |pages=774–775 |doi=10.1107/S0365110X64002006 |bibcode=1964AcCry..17..774O |issn=0365-110X}}</ref><ref>{{Cite journal |last1=Vikarchuk |first1=A.A. |last2=Gryzunova |first2=N.N. |last3=Gutkin |first3=M.Yu. |last4=Romanov |first4=A.E. |date=2018 |title=Copper Pentagonal Micropyramids Grown by Mechanically Activated Electrodeposition |url=https://www.degruyter.com/document/doi/10.1515/rams-2018-0030/html |journal=Reviews on Advanced Materials Science |language=en |volume=55 |issue=1 |pages=78–81 |doi=10.1515/rams-2018-0030 |issn=1605-8127}}</ref> or palladium<ref>{{Cite journal |last1=Fukaya |first1=Koji |last2=Ino |first2=Shozo |last3=Ogawa |first3=Shiro |date=1978 |title=Orientation and Structure of Palladium Particles Formed by Evaporation on Alkalihalide Crystals |url=https://www.jstage.jst.go.jp/article/matertrans1960/19/8/19_8_445/_article |journal=Transactions of the Japan Institute of Metals |language=en |volume=19 |issue=8 |pages=445–453 |doi=10.2320/matertrans1960.19.445 |issn=0021-4434}}</ref><ref>{{Cite journal |last1=Xiong |first1=Yujie |last2=Cai |first2=Honggang |last3=Yin |first3=Yadong |last4=Xia |first4=Younan |date=2007 |title=Synthesis and characterization of fivefold twinned nanorods and right bipyramids of palladium |url=https://linkinghub.elsevier.com/retrieve/pii/S0009261407005453 |journal=Chemical Physics Letters |language=en |volume=440 |issue=4–6 |pages=273–278 |doi=10.1016/j.cplett.2007.04.074|bibcode=2007CPL...440..273X }}</ref> and less often for some of the other [[face-centered cubic]] (fcc) metals such as nickel.<ref name=":7" /> There are also cases such as pentagonite where the crystal structure allows for five-fold twinning with minimal to no elastic strain  ([[#Disclination strain|see later]]).<ref name=":6" /> There is work where they have been observed in [[colloidal crystal]]s consisting of ordered arrays of nanoparticles,<ref>{{Cite journal |last1=Rupich |first1=Sara M. |last2=Shevchenko |first2=Elena V. |last3=Bodnarchuk |first3=Maryna I. |last4=Lee |first4=Byeongdu |last5=Talapin |first5=Dmitri V. |date=2010 |title=Size-Dependent Multiple Twinning in Nanocrystal Superlattices |url=https://pubs.acs.org/doi/10.1021/ja9074425 |journal=Journal of the American Chemical Society |language=en |volume=132 |issue=1 |pages=289–296 |doi=10.1021/ja9074425 |pmid=19968283 |bibcode=2010JAChS.132..289R |issn=0002-7863}}</ref><ref>{{Cite journal |last1=Lee |first1=Sangmin |last2=Glotzer |first2=Sharon C. |date=2022 |title=Entropically engineered formation of fivefold and icosahedral twinned clusters of colloidal shapes |journal=Nature Communications |language=en |volume=13 |issue=1 |page=7362 |doi=10.1038/s41467-022-34891-5 |issn=2041-1723 |pmc=9712591 |pmid=36450709|bibcode=2022NatCo..13.7362L }}</ref> and single crystals composed on individual decahedral nanoparticles.<ref>{{Cite journal |last1=Song |first1=Yongbo |last2=Li |first2=Yingwei |last3=Li |first3=Hao |last4=Ke |first4=Feng |last5=Xiang |first5=Ji |last6=Zhou |first6=Chuanjun |last7=Li |first7=Peng |last8=Zhu |first8=Manzhou |last9=Jin |first9=Rongchao |date=2020 |title=Atomically resolved Au52Cu72(SR)55 nanoalloy reveals Marks decahedron truncation and Penrose tiling surface |journal=Nature Communications |language=en |volume=11 |issue=1 |page=478 |doi=10.1038/s41467-020-14400-2 |issn=2041-1723 |pmc=6981204 |pmid=31980671}}</ref> There has been extensive modeling by many different approaches such as [[Embedded atom model|embedded atom]],<ref name=":7" /> many body,<ref name=":31">{{Cite journal |last1=Uppenbrink |first1=Julia |last2=Wales |first2=David J. |date=1992 |title=Structure and energetics of model metal clusters |url=https://pubs.aip.org/jcp/article/96/11/8520/855021/Structure-and-energetics-of-model-metal |journal=The Journal of Chemical Physics |language=en |volume=96 |issue=11 |pages=8520–8534 |doi=10.1063/1.462305 |bibcode=1992JChPh..96.8520U |issn=0021-9606}}</ref> [[molecular dynamics]],<ref name=":14">{{Cite journal |last1=Ascencio |first1=J.A. |last2=Pérez-Alvarez |first2=M. |last3=Tehuacanero |first3=S. |last4=José-Yacamán |first4=M. |date=2001 |title=Experimental and theoretical studies of instabilities of metal nanoparticles: a new kind of quasimelting |url=http://link.springer.com/10.1007/s003390100850 |journal=Applied Physics A: Materials Science & Processing |language=en |volume=73 |issue=3 |pages=295–300 |doi=10.1007/s003390100850 |bibcode=2001ApPhA..73..295A |issn=0947-8396}}</ref> [[tight binding]] approaches,<ref>{{Cite journal |last1=Gafner |first1=Yu. Ya. |last2=Gafner |first2=S. L. |last3=Chepkasov |first3=I. V. |date=2010 |title=The effect of thermal treatment on the organization of copper and nickel nanoclusters synthesized from the gas phase |url=http://link.springer.com/10.1134/S1063776110100110 |journal=Journal of Experimental and Theoretical Physics |language=en |volume=111 |issue=4 |pages=608–618 |doi=10.1134/S1063776110100110 |bibcode=2010JETP..111..608G |issn=1063-7761}}</ref> and [[density functional theory]] methods<ref>{{Cite journal |last1=Li |first1=Hui |last2=Li |first2=Lei |last3=Pedersen |first3=Andreas |last4=Gao |first4=Yi |last5=Khetrapal |first5=Navneet |last6=Jónsson |first6=Hannes |last7=Zeng |first7=Xiao Cheng |date=2015 |title=Magic-Number Gold Nanoclusters with Diameters from 1 to 3.5 nm: Relative Stability and Catalytic Activity for CO Oxidation |url=https://pubs.acs.org/doi/10.1021/nl504192u |journal=Nano Letters |language=en |volume=15 |issue=1 |pages=682–688 |doi=10.1021/nl504192u |pmid=25493586 |bibcode=2015NanoL..15..682L |issn=1530-6984}}</ref> as discussed by Francesca Baletto and Riccardo Ferrando<ref name=":16" /> and also discussed for [[#Landscape|energy landscapes]] later.

== Disclination strain ==
[[File:Quadaugmented tetrahedron.png|thumb|Pentagonal bipyramid showing the angular gap for [[face-centered cubic]].]]
These particles consist of five different (single crystal) units which are joined together by [[Grain boundary|twin boundaries]]. The simplest form shown in the figure has five [[Tetrahedron|tetrahedral]] crystals which most commonly have a [[face centered cubic]] structure, but there are other possibilities such as [[diamond cubic]] and a few others as well as more complex shapes. The angle between two twin planes is approximately 70.5 degrees in fcc, so five of these sums to 352.5 degrees (not 360 degrees) leading to an angular gap. At small sizes this gap is closed by an [[elastic deformation]], which Roland de Wit pointed out<ref name=":40">{{Cite journal |last=de Wit |first=R. |date=1971 |title=Relation between Dislocations and Disclinations |url=https://pubs.aip.org/jap/article/42/9/3304/504403/Relation-between-Dislocations-and-Disclinations |journal=Journal of Applied Physics |language=en |volume=42 |issue=9 |pages=3304–3308 |doi=10.1063/1.1660730 |bibcode=1971JAP....42.3304D |issn=0021-8979}}</ref><ref name=":3">{{Cite journal |last=Wit |first=R de |date=1972 |title=Partial disclinations |url=https://iopscience.iop.org/article/10.1088/0022-3719/5/5/004 |journal=Journal of Physics C: Solid State Physics |volume=5 |issue=5 |pages=529–534 |doi=10.1088/0022-3719/5/5/004 |bibcode=1972JPhC....5..529D |issn=0022-3719}}</ref> could be described as a wedge [[disclination]], a type of defect first discussed by [[Vito Volterra]] in 1907.<ref>{{Cite journal |last=Volterra |first=Vito |date=1907 |title=Sur l'équilibre des corps élastiques multiplement connexes |url=http://dx.doi.org/10.24033/asens.583 |journal=Annales scientifiques de l'École normale supérieure |volume=24 |pages=401–517 |doi=10.24033/asens.583 |issn=0012-9593}}</ref> With a disclination the strains to close the gap vary radially and are distributed throughout the particle.

With other structures the angle can be different; marcasite has a twin angle of 74.6 degrees, so instead of closing a missing wedge, one of angle 13 degrees has to be opened, which would be termed a negative disclination of 13 degrees. It has been pointed out by Chao Liang and Yi Yu<ref>{{Cite journal |last1=Liang |first1=Chao |last2=Yu |first2=Yi |date=2019 |title=Understanding the formation of multiply twinned structure in decahedral intermetallic nanoparticles |url=https://scripts.iucr.org/cgi-bin/paper?S2052252519002562 |journal=IUCrJ |volume=6 |issue=3 |pages=447–453 |doi=10.1107/S2052252519002562 |issn=2052-2525 |pmc=6503919 |pmid=31098025|bibcode=2019IUCrJ...6..447L }}</ref> that when [[Intermetallic particle|intermetallics]] are included there is a range of different angles, some similar to fcc where there is a deficiency (positive disclination), others such as AuCu where there is an overlap (negative disclination) similar to marcasite,<ref name=":25" /><ref>{{Cite journal |last=Arrouvel |first=Corinne |date=2021 |title=Surfaces, Interfaces and Crystal Growth of Marcasite FeS2 |journal=Materials Research |volume=24 |issue=1 |doi=10.1590/1980-5373-mr-2020-0383 |issn=1980-5373|doi-access=free }}</ref> while pentagonite has probably the smallest overlap at 3.5 degrees.<ref name=":6" />

{{anchor|Strain}}[[File:Von_Mises_Stress_Vertical.tif|left|thumb|Top view of Von Mises stress in pentagonal bipyramid and Marks decahedron.<ref name=":50" />]]
Early experimental [[high-resolution transmission electron microscopy]] data<ref name=":1" /> supported the idea of a distributed disclination strain field in the nanoparticles, as did dark field and other imaging modes in electron microscopes.<ref name=":49">{{Cite journal |last1=Marks |first1=L. D. |last2=Smith |first2=David J. |date=1983 |title=HREM and STEM of defects in multiply-twinned particles |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1365-2818.1983.tb04222.x |journal=Journal of Microscopy |language=en |volume=130 |issue=2 |pages=249–261 |doi=10.1111/j.1365-2818.1983.tb04222.x |issn=0022-2720}}</ref> In larger particles [[dislocation]]s have been detected to relieve some of the strain.<ref>{{Cite journal |last1=Nepijko |first1=S.A. |last2=Styopkin |first2=V.I. |last3=Hofmeister |first3=H. |last4=Scholtz |first4=R. |date=1986 |title=Defects in multiply-twinned particles |url=https://linkinghub.elsevier.com/retrieve/pii/0022024886903994 |journal=Journal of Crystal Growth |language=en |volume=76 |issue=2 |pages=501–506 |doi=10.1016/0022-0248(86)90399-4|bibcode=1986JCrGr..76..501N }}</ref><ref name=":30" /><ref>{{Cite journal |last=Hofmeister |first=H. |date=1991 |title=Lattice defects in decahedral multiply twinned particles of palladium |url=http://link.springer.com/10.1007/BF01448317 |journal=Zeitschrift für Physik D: Atoms, Molecules and Clusters |language=en |volume=19 |issue=1–4 |pages=307–310 |doi=10.1007/BF01448317 |bibcode=1991ZPhyD..19..307H |issn=0178-7683}}</ref><ref name=":20" /> The disclination deformation requires an energy which scales with the particle volume, so dislocations or [[grain boundaries]] are lower in energy for large sizes.<ref name=":5" />

More recently there has been detailed analysis of the atomic positions first by Craig Johnson et al,<ref name=":9">{{Cite journal |last1=Johnson |first1=Craig L. |last2=Snoeck |first2=Etienne |last3=Ezcurdia |first3=Manex |last4=Rodríguez-González |first4=Benito |last5=Pastoriza-Santos |first5=Isabel |last6=Liz-Marzán |first6=Luis M. |last7=Hÿtch |first7=Martin J. |date=2008 |title=Effects of elastic anisotropy on strain distributions in decahedral gold nanoparticles |url=https://www.nature.com/articles/nmat2083 |journal=Nature Materials |language=en |volume=7 |issue=2 |pages=120–124 |doi=10.1038/nmat2083 |pmid=18084296 |bibcode=2008NatMa...7..120J |issn=1476-1122}}</ref> followed up by a number of other authors,<ref name=":46" /><ref name=":48">{{Cite journal |last1=Ji |first1=Wenhai |last2=Qi |first2=Weihong |last3=Li |first3=Xu |last4=Zhao |first4=Shilei |last5=Tang |first5=Shasha |last6=Peng |first6=Hongcheng |last7=Li |first7=Siqi |date=2015 |title=Investigation of disclinations in Marks decahedral Pd nanoparticles by aberration-corrected HRTEM |url=https://linkinghub.elsevier.com/retrieve/pii/S0167577X15005315 |journal=Materials Letters |language=en |volume=152 |pages=283–286 |doi=10.1016/j.matlet.2015.03.137|bibcode=2015MatL..152..283J }}</ref><ref name=":10">{{Cite journal |last1=Wu |first1=Hao |last2=Yu |first2=Rong |last3=Zhu |first3=Jing |last4=Chen |first4=Wei |last5=Li |first5=Yadong |last6=Wang |first6=Tao |date=2021 |title=Size-dependent strain in fivefold twins of gold |url=https://scripts.iucr.org/cgi-bin/paper?S2052520620014791 |journal=Acta Crystallographica Section B Structural Science, Crystal Engineering and Materials |volume=77 |issue=1 |pages=93–98 |doi=10.1107/S2052520620014791 |issn=2052-5206}}</ref> providing more information on the strains and showing how they are distributed in the particles. While the classic disclination strain field is a reasonable first approximation model, there are differences when more complete elastic models are used such as [[Finite element method in structural mechanics|finite element methods]], particularly as pointed out by Johnson et al, [[Orthotropic material#Anisotropic elasticity|anisotropic elasticity]] needs to be used.<ref name=":9" /><ref name=":36" /><ref name=":10" /> One further complication is that the strain field is three dimensional, and more complex approaches are needed to measure the full details as detailed by Bart Goris et al, who also mention issues with strain from the support film.<ref>{{Cite journal |last1=Goris |first1=Bart |last2=De Beenhouwer |first2=Jan |last3=De Backer |first3=Annick |last4=Zanaga |first4=Daniele |last5=Batenburg |first5=K. Joost |last6=Sánchez-Iglesias |first6=Ana |last7=Liz-Marzán |first7=Luis M. |last8=Van Aert |first8=Sandra |last9=Bals |first9=Sara |last10=Sijbers |first10=Jan |last11=Van Tendeloo |first11=Gustaaf |date=2015 |title=Measuring Lattice Strain in Three Dimensions through Electron Microscopy |journal=Nano Letters |language=en |volume=15 |issue=10 |pages=6996–7001 |doi=10.1021/acs.nanolett.5b03008 |issn=1530-6984 |pmc=4877113 |pmid=26340328|bibcode=2015NanoL..15.6996G }}</ref> In addition, as pointed out by Srikanth Patala, [[Monica Olvera de la Cruz]] and Marks<ref name=":50">{{Cite journal |last1=Patala |first1=Srikanth |last2=Marks |first2=Laurence D. |last3=Olvera de la Cruz |first3=Monica |date=2013 |title=Elastic Strain Energy Effects in Faceted Decahedral Nanoparticles |url=https://pubs.acs.org/doi/10.1021/jp310045g |journal=The Journal of Physical Chemistry C |language=en |volume=117 |issue=3 |pages=1485–1494 |doi=10.1021/jp310045g |issn=1932-7447}}</ref> and shown in the figure, the [[Von Mises stress]] are different for (kinetic growth) pentagonal bipyramids versus the minimum energy shape.<ref name=":50" /> As of 2024 the strains are consistent with finite element calculations and a disclination strain field, with the possible addition of a shear component at the twin boundaries to accommodate some of the strains.<ref name=":9" /><ref name=":48" /><ref name=":10" />

An alternative to the disclination strain model which was proposed by B G Bagley in 1965 for whiskers<ref>{{Cite journal |last=Bagley |first=B. G. |date=1965 |title=A Dense Packing of Hard Spheres with Five-fold Symmetry |url=https://www.nature.com/articles/208674a0 |journal=Nature |language=en |volume=208 |issue=5011 |pages=674–675 |doi=10.1038/208674a0 |bibcode=1965Natur.208..674B |issn=0028-0836}}</ref> is that there is a change in the atomic structure away from face-centered cubic; a hypothesis that a [[Tetragonal crystal system|tetragonal crystal structure]]<ref name=":32" /> is lower in energy than fcc, and a lower energy atomic structure leads to the decahedral particles. This view was expanded upon by Cary Y. Yang<ref>{{Cite journal |last=Yang |first=C.Y. |date=1979 |title=Crystallography of decahedral and icosahedral particles |url=https://linkinghub.elsevier.com/retrieve/pii/0022024879902525 |journal=Journal of Crystal Growth |language=en |volume=47 |issue=2 |pages=274–282 |doi=10.1016/0022-0248(79)90252-5}}</ref> and can also be found in some of the early work of [[Miguel José Yacamán]].<ref>{{Cite journal |last1=Heinemann |first1=K. |last2=Yacamán |first2=M.J. |last3=Yang |first3=C.Y. |last4=Poppa |first4=H. |date=1979 |title=The structure of small, vapor-deposited particles |url=https://linkinghub.elsevier.com/retrieve/pii/0022024879902409 |journal=Journal of Crystal Growth |language=en |volume=47 |issue=2 |pages=177–186 |doi=10.1016/0022-0248(79)90240-9}}</ref><ref>{{Cite journal |last1=Yang |first1=C.Y. |last2=Heinemann |first2=K. |last3=Yacamán |first3=M.J. |last4=Poppa |first4=H. |date=1979 |title=A structural analysis of small vapor-deposited "multiply twinned" gold particles |url=https://linkinghub.elsevier.com/retrieve/pii/0040609079902311 |journal=Thin Solid Films |language=en |volume=58 |issue=1 |pages=163–168 |doi=10.1016/0040-6090(79)90231-1|bibcode=1979TSF....58..163Y }}</ref> There have been measurements of the average structure using [[X-ray diffraction]] which it has been argued support this view.<ref>{{Cite journal |last1=Sun |first1=Yugang |last2=Ren |first2=Yang |last3=Liu |first3=Yuzi |last4=Wen |first4=Jianguo |last5=Okasinski |first5=John S. |last6=Miller |first6=Dean J. |date=2012 |title=Ambient-stable tetragonal phase in silver nanostructures |journal=Nature Communications |language=en |volume=3 |issue=1 |page=971 |doi=10.1038/ncomms1963 |pmid=22828631 |bibcode=2012NatCo...3..971S |issn=2041-1723|doi-access=free }}</ref> However, these x-ray measurements only see the average which necessarily shows a tetragonal arrangement, and there is extensive evidence for inhomogeneous deformations dating back to the early work of Allpress and Sanders,<ref name=":0" /> Tsutomu Komoda,<ref name=":1" /> Marks and [[David J. Smith (physicist)|David J. Smith]]<ref name=":49" /> and more recently by high resolution imaging of details of the atomic structure.<ref name=":9" /><ref name=":46" /><ref name=":48" /><ref name=":10" /> As mentioned [[#Strain|above]], as of 2024 experimental imaging supports a disclination model with anisotropic elasticity.

== Three-dimensional shape ==
[[File:Different_Marks_Decahedra.gif|thumb|Decahedra for different (100) to (111) surface energies; top, down the common axis, and bottom from the side.<ref name=":22" />|260x260px]]
[[File:Au-Gold-2-1-16 Miass 2018-03-01 11-05-41 V3.gif|thumb|Gold fiveling, 0.5cm tall from Miass, Siberia, Russia, a Marks decahedron with <math>\gamma_{111} \approx 0.86\gamma_{100}</math>.]]The three-dimensional shape depends upon how the fivelings are formed, including the environment such as gas pressure and temperature. In the very early work only pentagonal bipyramids were reported.<ref name=":18" /><ref name=":19" /><ref name=":0" /> In 1970 Ino tried to model the energetics, but found that these bipyramids were higher in energy than single crystals with a [[Wulff construction]] shape. He found a lower energy form where he added {100} facets,<ref name=":15">{{Cite journal |last=Ino |first=Shozo |date=1969 |title=Stability of Multiply-Twinned Particles |url=https://journals.jps.jp/doi/10.1143/JPSJ.27.941 |journal=Journal of the Physical Society of Japan |language=en |volume=27 |issue=4 |pages=941–953 |doi=10.1143/JPSJ.27.941 |bibcode=1969JPSJ...27..941I |issn=0031-9015}}</ref> what is now commonly called the '''Ino decahedron'''. The surface energy of this form and a related [[Icosahedral twins|icosahedral twin]] scale as the two-thirds power of the volume, so they can be lower in energy than a single crystal as discussed further [[#Energy versus size|below]].

However, while Ino was able to explain the icosahedral particles, he was not able to explain the decahedral ones. Later [[Laurence D. Marks]] proposed a model using both experimental data and a theoretical analysis, which is based upon a [[Extended Wulff constructions#Modified Wulff construction|modified Wulff construction]] which includes more surface facets, including Ino's {100} as well as re-entrant {111} surfaces at the twin boundaries with the possibility of others such as {110}, while retaining the decahedral <math>D_{5h}</math> point group symmetry.<ref name=":4">{{Cite journal |last=Marks |first=L.D. |date=1983 |title=Modified Wulff constructions for twinned particles |url=https://linkinghub.elsevier.com/retrieve/pii/0022024883901847 |journal=Journal of Crystal Growth |language=en |volume=61 |issue=3 |pages=556–566 |doi=10.1016/0022-0248(83)90184-7|bibcode=1983JCrGr..61..556M }}</ref><ref name=":12" /><ref name=":5">{{Cite journal |last1=Howie |first1=A. |last2=Marks |first2=L. D. |date=1984 |title=Elastic strains and the energy balance for multiply twinned particles |url=http://www.tandfonline.com/doi/abs/10.1080/01418618408233432 |journal=Philosophical Magazine A |language=en |volume=49 |issue=1 |pages=95–109 |doi=10.1080/01418618408233432 |bibcode=1984PMagA..49...95H |issn=0141-8610}}</ref> This approach also includes the effect of gas and other environmental factors via how they change the [[surface energy]] of different facets. By combining this model with de Wit's elasticity,<ref name=":3" /> [[Archibald Howie]] and Marks were able to rationalize the stability of the decahedral to particles.<ref name=":5" /> Other work soon confirmed the shape reported by Marks for annealed particles.<ref>{{Cite journal |last1=Pérez-Ramírez |first1=J.G. |last2=José-Yacamán |first2=M. |last3=Díaz-Pérez |first3=Arturo |last4=Berriel-Valdós |first4=Raúl |date=1985 |title=On the equilibrium shape of multiple-twinned particles |url=https://linkinghub.elsevier.com/retrieve/pii/S0749603685800197 |journal=Superlattices and Microstructures |language=en |volume=1 |issue=6 |pages=485–487 |doi=10.1016/S0749-6036(85)80019-7|bibcode=1985SuMi....1..485P }}</ref> This was further confirmed in detailed atomistic calculations a few years later by Charles Cleveland and [[Uzi Landman]] who coined the term '''Marks decahedra''' for these shapes,<ref name=":7">{{Cite journal |last1=Cleveland |first1=Charles L. |last2=Landman |first2=Uzi |date=1991 |title=The energetics and structure of nickel clusters: Size dependence |url=https://pubs.aip.org/jcp/article/94/11/7376/96436/The-energetics-and-structure-of-nickel-clusters |journal=The Journal of Chemical Physics |language=en |volume=94 |issue=11 |pages=7376–7396 |doi=10.1063/1.460169 |bibcode=1991JChPh..94.7376C |issn=0021-9606}}</ref> this name now being widely used.<ref name=":26" /><ref name=":27" /><ref name=":21" /><ref name=":28" />

The minimum energy or thermodynamic shape for these particles<ref name=":4" /><ref name=":12" /> depends upon the relative surface energies of different facets, similar to a single crystal Wulff shape; they are formed by combining segments of a conventional Wulff construction with two additional internal facets to represent the twin boundaries.<ref name=":12" /><ref name=":4" /> An overview of codes to calculate these shapes was published in 2021 by Christina Boukouvala et al.<ref name=":37">{{Cite journal |last1=Boukouvala |first1=Christina |last2=Daniel |first2=Joshua |last3=Ringe |first3=Emilie |date=2021 |title=Approaches to modelling the shape of nanocrystals |journal=Nano Convergence |language=en |volume=8 |issue=1 |page=26 |doi=10.1186/s40580-021-00275-6 |doi-access=free |issn=2196-5404 |pmc=8429535 |pmid=34499259|bibcode=2021NanoC...8...26B }}</ref> Considering just {111} and {100} facets:<ref name=":4" /><ref name=":12" />

* The Ino decahedron occurs when the surface energy of the {100} facets is small, <math>\gamma_{111} > 2 \gamma_{100}/\sqrt{3}</math>; 
* Common is the Marks decahedron with {100} facets and a re-entrant surface at the twin boundaries for <math>\gamma_{100}/\sqrt{3} < \gamma_{111} < 2\gamma_{100}/\sqrt{3}</math>
* With <math>\gamma_{111} < \gamma_{100}/\sqrt{3}</math> there is no {100} faceting, and the particles have been called nanostars.<ref>{{Cite journal |last1=Jin |first1=Biao |last2=Yan |first2=Feng |last3=Qi |first3=Xin |last4=Cai |first4=Bin |last5=Tao |first5=Jinhui |last6=Fu |first6=Xiaofeng |last7=Tan |first7=Susheng |last8=Zhang |first8=Peijun |last9=Pfaendtner |first9=Jim |last10=Naser |first10=Nada Y. |last11=Baneyx |first11=François |last12=Zhang |first12=Xin |last13=DeYoreo |first13=James J. |last14=Chen |first14=Chun-Long |date=2022 |title=Peptoid-Directed Formation of Five-Fold Twinned Au Nanostars through Particle Attachment and Facet Stabilization |journal=Angewandte Chemie International Edition |language=en |volume=61 |issue=14 |pages=e202201980 |doi=10.1002/anie.202201980 |issn=1433-7851 |pmc=9258440 |pmid=35167709}}</ref>
* For very low <math>\gamma_{100}</math> the equilibrium shape is a long rod along the common five-fold axis.

The photograph of an 0.5&nbsp;cm gold fiveling from [[Miass]] is a Marks decahedron with <math>\gamma_{111} \approx 0.85 \gamma_{100}</math>, while the sketch of Rose<ref name=":11" /> is for <math>\gamma_{111} \approx 0.7 \gamma_{100}</math>.  The [[#LJ75|75 atom]] cluster shown above corresponds to the same shape for a small number of atoms. Experimentally, in fcc crystals fivelings with only {111} and {100} facets are common, but many other facets can be present in the Wulff construction leading to more rounded shapes,<ref name=":12" /><ref name=":21">{{Cite journal |last1=Casillas hi |first1=Gilberto |last2=Velázquez-Salazar |first2=J. Jesús |last3=Jose-Yacaman |first3=Miguel |date=2012 |title=A New Mechanism of Stabilization of Large Decahedral Nanoparticles |journal=The Journal of Physical Chemistry C |language=en |volume=116 |issue=15 |pages=8844–8848 |doi=10.1021/jp3011475 |issn=1932-7447 |pmc=3353654 |pmid=22609961}}</ref> for instance {113} facets for silicon.<ref name=":39">{{Cite journal |last1=Takeguchi |first1=Masaki |last2=Tanaka |first2=Miyoko |last3=Yasuda |first3=Hidehiro |last4=Furuya |first4=Kazuo |date=2001 |title=Real-time high-resolution transmission electron microscopy observation of the growth process of ( 001 ) surfaces on a nanometer-sized Si multiply twinned particle |url=https://linkinghub.elsevier.com/retrieve/pii/S003960280101247X |journal=Surface Science |language=en |volume=493 |issue=1–3 |pages=414–419 |doi=10.1016/S0039-6028(01)01247-X|bibcode=2001SurSc.493..414T }}</ref> It is known that the surface can [[Surface reconstruction|reconstruct]] to a different atomic arrangement in the outermost atomic plane, for instance a dimer reconstruction for {100} facets of silicon particles<ref name=":39" /> of a hexagonal overlayer on the {100} facets of gold decahedra.<ref name=":21" />
[[File:5fling-Rod.jpg|left|thumb|SEM image of decagonal rod of silver.<ref name="linkinghub.elsevier.com">{{Cite journal |last1=Reyes-Gasga |first1=J. |last2=Elechiguerra |first2=J.L. |last3=Liu |first3=C. |last4=Camacho-Bragado |first4=A. |last5=Montejano-Carrizales |first5=J.M. |last6=Jose Yacaman |first6=M. |date=2006 |title=On the structure of nanorods and nanowires with pentagonal cross-sections |url=https://linkinghub.elsevier.com/retrieve/pii/S002202480501081X |journal=Journal of Crystal Growth |language=en |volume=286 |issue=1 |pages=162–172 |doi=10.1016/j.jcrysgro.2005.09.028|bibcode=2006JCrGr.286..162R }}</ref>]]
What shape is present depends not just on the surface energy of the different facets, but also upon how the particles grow. The thermodynamic shape is determined by the [[Wulff construction]], which considers the energy of each possible surface facet and yields the lowest energy shape. The original Marks decahedron was based upon a form of Wulff construction that takes into account the twin boundaries.<ref name=":4" /><ref name=":12">{{Cite journal |last=Marks |first=L. D. |date=1984 |title=Surface structure and energetics of multiply twinned particles |url=http://www.tandfonline.com/doi/abs/10.1080/01418618408233431 |journal=Philosophical Magazine A |language=en |volume=49 |issue=1 |pages=81–93 |doi=10.1080/01418618408233431 |bibcode=1984PMagA..49...81M |issn=0141-8610}}</ref> There is a related [[Extended Wulff constructions#Kinetic Wulff construction|kinetic Wulff construction]] where the growth rate of different surfaces is used instead of the energies.<ref name=":22">{{Cite journal |last1=Ringe |first1=Emilie |last2=Van Duyne |first2=Richard P. |last3=Marks |first3=Laurence D. |date=2013 |title=Kinetic and Thermodynamic Modified Wulff Constructions for Twinned Nanoparticles |url=https://pubs.acs.org/doi/10.1021/jp401566m |journal=The Journal of Physical Chemistry C |language=en |volume=117 |issue=31 |pages=15859–15870 |doi=10.1021/jp401566m |issn=1932-7447}}</ref><ref name=":35">{{Cite journal |last1=Li |first1=B. |last2=Lowengrub |first2=J. |last3=Ratz |first3=A. |last4=Voigt |first4=A. |date=2009 |title=Geometric Evolution Laws for Thin Crystalline Films: Modeling and Numerics |url=https://global-sci.org/intro/article_detail/cicp/7688.html |journal=Communications in Computational Physics |volume=6 |issue=3 |pages=433–482}}</ref> This type of growth matters when the formation of a new [[Island growth|island]] on a flat facet limits the growth rate.<ref>{{Cite journal |last1=Combe |first1=Nicolas |last2=Jensen |first2=Pablo |last3=Pimpinelli |first3=Alberto |date=2000 |title=Changing Shapes in the Nanoworld |url=https://link.aps.org/doi/10.1103/PhysRevLett.85.110 |journal=Physical Review Letters |language=en |volume=85 |issue=1 |pages=110–113 |doi=10.1103/PhysRevLett.85.110 |pmid=10991171 |arxiv=cond-mat/0005113 |bibcode=2000PhRvL..85..110C |issn=0031-9007}}</ref> If the {100} surfaces of Ino grow faster, then they will not appear in the final shape, similarly for the re-entrant surfaces at the twin boundaries—this leads to the pentagonal bipyramids often observed.<ref name=":22" /> Alternatively, if the {111} surfaces grow fast and {100} slow the kinetic shape will be a long rod along the common five-fold axis as shown in the figure.<ref>{{Cite journal |last1=Ni |first1=Chaoying |last2=Hassan |first2=Puthusserickal A. |last3=Kaler |first3=Eric W. |date=2005 |title=Structural Characteristics and Growth of Pentagonal Silver Nanorods Prepared by a Surfactant Method |url=https://pubs.acs.org/doi/10.1021/la046807c |journal=Langmuir |language=en |volume=21 |issue=8 |pages=3334–3337 |doi=10.1021/la046807c |pmid=15807571 |issn=0743-7463}}</ref><ref>{{Cite journal |last1=Wang |first1=Jen-Hung |last2=Yang |first2=Tze-Hsien |last3=Wu |first3=Wen-Wei |last4=Chen |first4=Lih-Juann |last5=Chen |first5=Chih-Hung |last6=Chu |first6=Cheng-Jie |date=2006 |title=Synthesis and growth mechanism of pentagonal Cu nanobats with field emission characteristics |url=https://iopscience.iop.org/article/10.1088/0957-4484/17/3/017 |journal=Nanotechnology |volume=17 |issue=3 |pages=719–722 |doi=10.1088/0957-4484/17/3/017 |bibcode=2006Nanot..17..719W |issn=0957-4484}}</ref><ref name="linkinghub.elsevier.com"/><ref>{{Cite journal |last1=Qi |first1=Xin |last2=Chen |first2=Zihao |last3=Yan |first3=Tianyu |last4=Fichthorn |first4=Kristen A. |date=2019 |title=Growth Mechanism of Five-Fold Twinned Ag Nanowires from Multiscale Theory and Simulations |url=https://pubs.acs.org/doi/10.1021/acsnano.9b00820 |journal=ACS Nano |language=en |volume=13 |issue=4 |pages=4647–4656 |doi=10.1021/acsnano.9b00820 |pmid=30869861 |osti=1594111 |issn=1936-0851}}</ref>
[[File:5ling-star.jpg|thumb|150x150px|Fiveling (decahedral nanoparticle) showing diffusion growth at tips.<ref name=":43" />]]
Another different set of shapes can occur when [[diffusion]] of atoms to the particles dominates, a growth regime called [[Crystal growth#Diffusion-control|diffusion controlled growth]]. In such cases surface [[curvature]] can play a major role,<ref>{{Cite journal |last1=Ball |first1=R. C. |last2=Blunt |first2=M. J. |last3=Rath Spivack |first3=O. |date=1989 |title=Diffusion-controlled growth |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1989.0045 |journal=Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences |language=en |volume=423 |issue=1864 |pages=123–132 |doi=10.1098/rspa.1989.0045 |bibcode=1989RSPSA.423..123B |issn=0080-4630}}</ref><ref name=":35" /> for instance leading to spikes originating at the sharp corners of a pentagonal bipyramids, sometimes leading to pointy stars, as shown in the figure.<ref name=":43">{{Cite journal |last1=Bazán-Díaz |first1=Lourdes |last2=Mendoza-Cruz |first2=Rubén |last3=Velázquez-Salazar |first3=J. Jesús |last4=Plascencia-Villa |first4=Germán |last5=Romeu |first5=David |last6=Reyes-Gasga |first6=José |last7=Herrera-Becerra |first7=Raúl |last8=José-Yacamán |first8=Miguel |last9=Guisbiers |first9=Grégory |date=2015 |title=Gold–copper nanostars as photo-thermal agents: synthesis and advanced electron microscopy characterization |url=http://xlink.rsc.org/?DOI=C5NR06491K |journal=Nanoscale |language=en |volume=7 |issue=48 |pages=20734–20742 |doi=10.1039/C5NR06491K |pmid=26602429 |bibcode=2015Nanos...720734B |issn=2040-3364}}</ref>

== Energy versus size ==
The most common approach to understand the formation of these particles, first used by Ino in 1969,<ref name=":15" /> is to look at the energy as a function of size comparing [[icosahedral twins]], decahedral nanoparticles and single crystals. The total energy for each type of particle can be written as the sum of three terms:

:<math>E_{total} = E_{surface} V^{2/3} + E_{strain} V + E_{surface\ stress}V^{2/3}</math>

for a volume <math>V</math>, where <math>E_{surface}</math> is the [[surface energy]], <math>E_{strain}</math> is the disclination [[strain energy]] to close the gap (or overlap for marcasite and others), and <math>E_{surface \ stress}</math> is a coupling term for the effect of the strain on the surface energy via the [[surface stress]],<ref>{{Cite journal |last1=Vermaak |first1=J.S. |last2=Mays |first2=C.W. |last3=Kuhlmann-Wilsdorf |first3=D. |date=1968 |title=On surface stress and surface tension |url=https://linkinghub.elsevier.com/retrieve/pii/0039602868901180 |journal=Surface Science |language=en |volume=12 |issue=2 |pages=128–133 |doi=10.1016/0039-6028(68)90118-0}}</ref><ref>{{Cite journal |last1=Mays |first1=C.W. |last2=Vermaak |first2=J.S. |last3=Kuhlmann-Wilsdorf |first3=D. |date=1968 |title=On surface stress and surface tension |url=https://linkinghub.elsevier.com/retrieve/pii/0039602868901192 |journal=Surface Science |language=en |volume=12 |issue=2 |pages=134–140 |doi=10.1016/0039-6028(68)90119-2|bibcode=1968SurSc..12..134M }}</ref><ref>{{Cite journal |last1=Müller |first1=Pierre |last2=Saùl |first2=Andres |last3=Leroy |first3=Frédéric |date=2013 |title=Simple views on surface stress and surface energy concepts |journal=Advances in Natural Sciences: Nanoscience and Nanotechnology |volume=5 |issue=1 |pages=013002 |doi=10.1088/2043-6262/5/1/013002 |issn=2043-6262|doi-access=free }}</ref> which can be a significant contribution.<ref name=":36">{{Cite journal |last1=Patala |first1=Srikanth |last2=Marks |first2=Laurence D. |last3=Olvera de la Cruz |first3=Monica |date=2013 |title=Thermodynamic Analysis of Multiply Twinned Particles: Surface Stress Effects |url=https://pubs.acs.org/doi/10.1021/jz401496d |journal=The Journal of Physical Chemistry Letters |language=en |volume=4 |issue=18 |pages=3089–3094 |doi=10.1021/jz401496d |issn=1948-7185}}</ref> The sum of these three terms is compared to the total surface energy of a single crystal (which has no strain), and to similar terms for an icosahedral particle. Because the decahedral particles have a lower total surface energy than single crystals due (approximately, in fcc) to more low energy {111} surfaces, they are lower in total energy for an intermediate size regime, with the icosahedral particles more stable at very small sizes. (The icosahedral particle have even more {111} surfaces, but also more strain.<ref name=":5" />) At large sizes the strain energy can become very large, so it is energetically favorable to have [[dislocation]]s and/or a [[grain boundary]] instead of a distributed strain.<ref name=":20">{{Cite journal |last1=Romanov |first1=Alexey E. |last2=Vikarchuk |first2=Anatoly A. |last3=Kolesnikova |first3=Anna L. |last4=Dorogin |first4=Leonid M. |last5=Kink |first5=Ilmar |last6=Aifantis |first6=Elias C. |date=2012 |title=Structural transformations in nano- and microobjects triggered by disclinations |url=http://link.springer.com/10.1557/jmr.2011.372 |journal=Journal of Materials Research |language=en |volume=27 |issue=3 |pages=545–551 |doi=10.1557/jmr.2011.372 |bibcode=2012JMatR..27..545R |issn=0884-2914}}</ref> The very large mineral samples are almost certainly trapped in metastable higher energy configurations.

{{anchor|Landscape}}There is no general consensus on the exact sizes when there is a transition in which type of particle is lowest in energy, as these vary with material and also the environment such as gas and temperature; the coupling surface stress term and also the surface energies of the facets are very sensitive to these.<ref>{{Cite journal |last=Feibelman |first=Peter J. |date=1997 |title=First-principles calculations of stress induced by gas adsorption on Pt(111) |url=https://link.aps.org/doi/10.1103/PhysRevB.56.2175 |journal=Physical Review B |language=en |volume=56 |issue=4 |pages=2175–2182 |doi=10.1103/PhysRevB.56.2175 |bibcode=1997PhRvB..56.2175F |issn=0163-1829}}</ref><ref>{{Cite journal |last1=Graoui |first1=H. |last2=Giorgio |first2=S. |last3=Henry |first3=C.R. |date=1998 |title=Shape variations of Pd particles under oxygen adsorption |url=https://linkinghub.elsevier.com/retrieve/pii/S0039602898006888 |journal=Surface Science |language=en |volume=417 |issue=2–3 |pages=350–360 |doi=10.1016/S0039-6028(98)00688-8|bibcode=1998SurSc.417..350G }}</ref><ref>{{Cite journal |last1=Wynblatt |first1=P. |last2=Chatain |first2=D. |date=2009 |title=Surface segregation anisotropy and the equilibrium crystal shape of alloy crystals |journal=Reviews on Advanced Materials Science |volume=21 |pages=44–56|s2cid=137869647 }}</ref> In addition, as first described by Michael Hoare and P Pal<ref name=":29">{{Cite journal |last1=Hoare |first1=M.R. |last2=Pal |first2=P. |date=1971 |title=Physical cluster mechanics: Statics and energy surfaces for monatomic systems |url=http://www.tandfonline.com/doi/abs/10.1080/00018737100101231 |journal=Advances in Physics |language=en |volume=20 |issue=84 |pages=161–196 |doi=10.1080/00018737100101231 |bibcode=1971AdPhy..20..161H |issn=0001-8732}}</ref> and [[R. Stephen Berry]]<ref name=":23">{{Cite journal |last1=Berry |first1=R. Stephen |last2=Jellinek |first2=Julius |last3=Natanson |first3=Grigory |date=1984 |title=Melting of clusters and melting |url=https://link.aps.org/doi/10.1103/PhysRevA.30.919 |journal=Physical Review A |language=en |volume=30 |issue=2 |pages=919–931 |doi=10.1103/PhysRevA.30.919 |bibcode=1984PhRvA..30..919B |issn=0556-2791}}</ref><ref name=":24">{{Cite journal |last=Berry |first=R. Stephen. |date=1993 |title=Potential surfaces and dynamics: what clusters tell us |url=https://pubs.acs.org/doi/abs/10.1021/cr00023a003 |journal=Chemical Reviews |language=en |volume=93 |issue=7 |pages=2379–2394 |doi=10.1021/cr00023a003 |issn=0009-2665}}</ref> and analyzed for these particles by [[Pulickel Ajayan]] and Marks<ref name=":8">{{Cite journal |last1=Ajayan |first1=P. M. |last2=Marks |first2=L. D. |date=1988 |title=Quasimelting and phases of small particles |url=https://link.aps.org/doi/10.1103/PhysRevLett.60.585 |journal=Physical Review Letters |language=en |volume=60 |issue=7 |pages=585–587 |doi=10.1103/PhysRevLett.60.585 |pmid=10038590 |bibcode=1988PhRvL..60..585A |issn=0031-9007}}</ref> as well as discussed by others such as [[Amanda Barnard]],<ref>{{Cite journal |last1=Barnard |first1=Amanda S. |last2=Young |first2=Neil P. |last3=Kirkland |first3=Angus I. |last4=van Huis |first4=Marijn A. |last5=Xu |first5=Huifang |date=2009 |title=Nanogold: A Quantitative Phase Map |url=https://pubs.acs.org/doi/10.1021/nn900220k |journal=ACS Nano |language=en |volume=3 |issue=6 |pages=1431–1436 |doi=10.1021/nn900220k |pmid=19489558 |issn=1936-0851}}</ref> [[David J. Wales]],<ref name=":31" /><ref name=":32">{{Cite book |last1=Wales |first1=David J. |url=https://onlinelibrary.wiley.com/doi/10.1002/9780470141748.ch1 |title=Energy Landscapes: From Clusters to Biomolecules |last2=Doye |first2=Jonathan P. K. |last3=Miller |first3=Mark A. |last4=Mortenson |first4=Paul N. |last5=Walsh |first5=Tiffany R. |date=2000 |publisher=Wiley |isbn=978-0-471-39331-3 |editor-last=Prigogine |editor-first=I. |edition=1 |volume=115 |pages=39–46 |language=en |doi=10.1002/9780470141748.ch1 |access-date=2024-04-01 |editor2-last=Rice |editor2-first=Stuart A.}}</ref><ref name=":33">{{Cite journal |last=Wales |first=David J. |date=2018 |title=Exploring Energy Landscapes |url=https://www.annualreviews.org/doi/10.1146/annurev-physchem-050317-021219 |journal=Annual Review of Physical Chemistry |language=en |volume=69 |issue=1 |pages=401–425 |bibcode=2018ARPC...69..401W |doi=10.1146/annurev-physchem-050317-021219 |issn=0066-426X |pmid=29677468}}</ref> [[Kristen Fichthorn]]<ref>{{Cite journal |last1=Zhang |first1=Huaizhong |last2=Khan |first2=Mohd Ahmed |last3=Yan |first3=Tianyu |last4=Fichthorn |first4=Kristen A. |date=2024 |title=Size and temperature dependent shapes of copper nanocrystals using parallel tempering molecular dynamics |journal=Nanoscale |volume=16 |issue=23 |pages=11146–11155 |language=en |doi=10.1039/D4NR00317A |pmid=38506642 |issn=2040-3364|doi-access=free }}</ref> and Baletto and Ferrando,<ref name=":16">{{Cite journal |last1=Mottet |first1=C. |last2=Goniakowski |first2=J. |last3=Baletto |first3=F. |last4=Ferrando |first4=R. |last5=Treglia |first5=G. |date=2004 |title=Modeling free and supported metallic nanoclusters: structure and dynamics |url=http://www.tandfonline.com/doi/abs/10.1080/1411590310001622473 |journal=Phase Transitions |language=en |volume=77 |issue=1–2 |pages=101–113 |doi=10.1080/1411590310001622473 |bibcode=2004PhaTr..77..101M |issn=0141-1594}}</ref> at very small sizes there will be a statistical population of different structures so many different ones will coexist. In many cases nanoparticles are believed to grow from a very small seed without changing shape, and reflect the distribution of coexisting structures.<ref name=":38" />  [[File:Energy_landscape_for_75_atom_Leonard-Jones_cluster.tif|thumb|Energy landscape for a 75 atom Leonard-Jones cluster for temperature and an order parameter.<ref name=":51" />]]For systems where icosahedral and decahedral morphologies are both relatively low in energy, the competition between these structures has implications for structure prediction and for the global thermodynamic and kinetic properties. These result from a double funnel energy landscape<ref>{{Cite book |last=Wales |first=David |url=https://www.cambridge.org/core/product/identifier/9780511721724/type/book |title=Energy Landscapes: Applications to Clusters, Biomolecules and Glasses |date=2001 |publisher=Cambridge University Press |isbn=978-0-521-81415-7 |edition=1 |pages=4590479 |doi=10.1017/cbo9780511721724}}</ref><ref>{{Cite journal |last1=Wales |first1=David J. |last2=Miller |first2=Mark A. |last3=Walsh |first3=Tiffany R. |date=1998 |title=Archetypal energy landscapes |url=https://www.nature.com/articles/29487 |journal=Nature |language=en |volume=394 |issue=6695 |pages=758–760 |doi=10.1038/29487 |bibcode=1998Natur.394..758W |issn=0028-0836}}</ref> where the two families of structures are separated by a relatively high energy barrier at the temperature where they are in [[thermodynamic equilibrium]]. This situation arises for a cluster of 75 atoms with the [[Lennard-Jones potential]], where the global [[potential energy]] minimum is decahedral, and structures based upon incomplete [[Alan Lindsay Mackay|Mackay]] icosahedra<ref>{{Cite journal |last=Mackay |first=A. L. |date=1962 |title=A dense non-crystallographic packing of equal spheres |url=https://scripts.iucr.org/cgi-bin/paper?S0365110X6200239X |journal=Acta Crystallographica |volume=15 |issue=9 |pages=916–918 |doi=10.1107/S0365110X6200239X |bibcode=1962AcCry..15..916M |issn=0365-110X}}</ref> are also low in potential energy, but higher in entropy. The free energy barrier between these families is large compared to the available thermal energy at the temperature where they are in equilibrium. An example is shown in the figure, with probability in the lower part and energy above with axes of an [[order parameter]] <math>Q_6</math> and temperature <math>T</math>. At low temperature the [[Fiveling#LJ75|75 atom]] decahedral cluster (Dh) is the global free energy minimum, but as the temperature increases the higher [[entropy]] of the competing structures based on incomplete icosahedra (Ic) causes the finite system analogue of a [[first-order phase transition]]; at even higher temperatures a liquid-like state is favored.<ref name=":51">{{Cite journal |last=Wales |first=David J. |date=2013 |title=Surveying a complex potential energy landscape: Overcoming broken ergodicity using basin-sampling |url=https://linkinghub.elsevier.com/retrieve/pii/S0009261413009573 |journal=Chemical Physics Letters |language=en |volume=584 |pages=1–9 |doi=10.1016/j.cplett.2013.07.066|bibcode=2013CPL...584....1W }}</ref>

There has been experiment support based upon work where single nanoparticles are imaged using electron microscopes either as they grow or as a function of time. One of the earliest works was that of Yagi et al<ref name=":2" /> who directly observed changes in the internal structure with time during growth. More recent work has observed variations in the internal structure in liquid cells,<ref name=":13" /> or changes between different forms due to either (or both) heating or the electron beam in an electron microscope<ref>{{Cite journal |last1=Iijima |first1=Sumio |last2=Ichihashi |first2=Toshinari |date=1986 |title=Structural instability of ultrafine particles of metals |url=https://link.aps.org/doi/10.1103/PhysRevLett.56.616 |journal=Physical Review Letters |language=en |volume=56 |issue=6 |pages=616–619 |doi=10.1103/PhysRevLett.56.616 |pmid=10033240 |bibcode=1986PhRvL..56..616I |issn=0031-9007}}</ref><ref>{{Cite journal |last1=Smith |first1=David J. |last2=Petford-Long |first2=Amanda K. |last3=Wallenberg |first3=L. R. |last4=Bovin |first4=J.-O. |date=1986 |title=Dynamic Atomic-Level Rearrangements in Small Gold Particles |url=https://www.science.org/doi/10.1126/science.233.4766.872 |journal=Science |language=en |volume=233 |issue=4766 |pages=872–875 |doi=10.1126/science.233.4766.872 |pmid=17752214 |bibcode=1986Sci...233..872S |issn=0036-8075}}</ref><ref>{{Cite journal |last1=Young |first1=N.P. |last2=van Huis |first2=M.A. |last3=Zandbergen |first3=H.W. |last4=Xu |first4=H. |last5=Kirkland |first5=A.I. |date=2010 |title=Transformations of gold nanoparticles investigated using variable temperature high-resolution transmission electron microscopy |url=https://linkinghub.elsevier.com/retrieve/pii/S0304399109002885 |journal=Ultramicroscopy |language=en |volume=110 |issue=5 |pages=506–516 |doi=10.1016/j.ultramic.2009.12.010|pmid=20083353 }}</ref> including substrate effects.<ref name=":14" />

== Successive twinning ==
Allpress and Sanders proposed an alternative approach to energy minimization to understanding these particles called "successive twinning".<ref name=":0" /> Here one starts with a single tetrahedral unit, which then forms a twin either by accident during growth or by collision with another tetrahedron. It was proposed that this could continue to eventually have five units join.<ref name=":34" />{{anchor|DMove}}
[[File:Disclination_movement_during_growth.gif|thumb|Atomistic simulation of disclination movement in decahedral particles, showing the energies relative to perfect Marks decahedra and tetrahedra.<ref name=":34" />|362x362px]]
The term "successive twinning" has now come to mean a related concept: motion of the disclination either to or from a symmetric position as sketched in the atomistic simulation in the figure;<ref name=":34" /> see also Haiqiang Zhao et al<ref name=":28" /> for very similar experimental images.

While in many cases experimental images show symmetric structures, sometimes they are less so and the five-fold center is quite asymmetric.<ref>{{Cite journal |last1=Uppenbrink |first1=J. |last2=Wales |first2=D. J. |last3=Kirkland |first3=A. I. |last4=Jefferson |first4=D. A. |last5=Urban |first5=J. |date=1992 |title=Structure and energetics of model symmetric and asymmetric decahedra |url=https://www.tandfonline.com/doi/full/10.1080/13642819208217922 |journal=Philosophical Magazine B |language=en |volume=65 |issue=5 |pages=1079–1096 |doi=10.1080/13642819208217922 |bibcode=1992PMagB..65.1079U |issn=1364-2812}}</ref><ref name=":28">{{Cite journal |last1=Zhao |first1=Haiqiang |last2=Qi |first2=Weihong |last3=Ji |first3=Wenhai |last4=Wang |first4=Tianran |last5=Peng |first5=Hongcheng |last6=Wang |first6=Qi |last7=Jia |first7=Yanlin |last8=He |first8=Jieting |date=2017 |title=Large Marks-decahedral Pd nanoparticles synthesized by a modified hydrothermal method using a homogeneous reactor |url=http://link.springer.com/10.1007/s11051-017-3856-0 |journal=Journal of Nanoparticle Research |language=en |volume=19 |issue=5 |page=162 |doi=10.1007/s11051-017-3856-0 |bibcode=2017JNR....19..162Z |issn=1388-0764}}</ref> There are asymmetric cases which can be metastable,<ref name=":4" /> and asymmetry can also be a strain relief process<ref>{{Cite journal |last1=Dundurs |first1=J. |last2=Marks |first2=L. D. |last3=Ajayan |first3=P. M. |date=1988 |title=Structural fluctuations in small particles |url=http://www.tandfonline.com/doi/abs/10.1080/01418618808214410 |journal=Philosophical Magazine A |language=en |volume=57 |issue=4 |pages=605–620 |doi=10.1080/01418618808214410 |bibcode=1988PMagA..57..605D |issn=0141-8610}}</ref> or involved in how the particle convert to single crystals or from single crystals.<ref name=":2" /><ref name=":8" />  During growth there may be changes, as directly observed by Katsumichi Yagi et al for growth inside an electron microscope,<ref name=":2">{{Cite journal |last1=Yagi |first1=K. |last2=Takayanagi |first2=K. |last3=Kobayashi |first3=K. |last4=Honjo |first4=G. |date=1975 |title=In-situ observations of growth processes of multiply twinned particles |url=https://linkinghub.elsevier.com/retrieve/pii/0022024875900330 |journal=Journal of Crystal Growth |language=en |volume=28 |issue=1 |pages=117–124 |doi=10.1016/0022-0248(75)90033-0|bibcode=1975JCrGr..28..117Y }}</ref>  and migration of the disclination from the outside has been observed in liquid-cell studies in electron microscopes.<ref name=":13">{{Cite journal |last1=Ma |first1=Xiaoming |last2=Lin |first2=Fang |last3=Chen |first3=Xin |last4=Jin |first4=Chuanhong |date=2020-08-25 |title=Unveiling Growth Pathways of Multiply Twinned Gold Nanoparticles by In Situ Liquid Cell Transmission Electron Microscopy |url=https://pubs.acs.org/doi/10.1021/acsnano.9b10173 |journal=ACS Nano |language=en |volume=14 |issue=8 |pages=9594–9604 |doi=10.1021/acsnano.9b10173 |pmid=32806061 |issn=1936-0851}}</ref> Extensive details about the atomic processes involved in motion of the disclination have been given using [[Molecular dynamics|molecular dynamics calculations]] supported by [[density functional theory]] as shown in the figure.<ref name=":34">{{Cite journal |last1=El Koraychy |first1=El Yakout |last2=Roncaglia |first2=Cesare |last3=Nelli |first3=Diana |last4=Cerbelaud |first4=Manuella |last5=Ferrando |first5=Riccardo |date=2022 |title=Growth mechanisms from tetrahedral seeds to multiply twinned Au nanoparticles revealed by atomistic simulations |url=http://xlink.rsc.org/?DOI=D1NH00599E |journal=Nanoscale Horizons |language=en |volume=7 |issue=8 |pages=883–889 |bibcode=2022NanoH...7..883E |doi=10.1039/D1NH00599E |issn=2055-6756 |pmid=35722927}}</ref>

== Connections ==
There are a number of related concepts and applications of decahedral particles.

=== Quasicrystals ===
Soon after the discovery of [[quasicrystal]]s it was suggested by [[Linus Pauling]]<ref>{{Cite journal |last=Pauling |first=Linus |date=1985 |title=Apparent icosahedral symmetry is due to directed multiple twinning of cubic crystals |url=https://www.nature.com/articles/317512a0 |journal=Nature |language=en |volume=317 |issue=6037 |pages=512–514 |doi=10.1038/317512a0 |bibcode=1985Natur.317..512P |issn=0028-0836}}</ref><ref>{{Cite journal |last=Pauling |first=Linus |date=1987 |title=So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal |url=https://link.aps.org/doi/10.1103/PhysRevLett.58.365 |journal=Physical Review Letters |volume=58 |issue=4 |pages=365–368 |doi=10.1103/PhysRevLett.58.365|pmid=10034915 |bibcode=1987PhRvL..58..365P }}</ref> that five-fold cyclic twins such as these were the source of the electron diffraction data observed by [[Dan Shechtman]].<ref>{{Cite journal |last1=Shechtman |first1=D. |last2=Blech |first2=I. |last3=Gratias |first3=D. |last4=Cahn |first4=J. W. |date=1984 |title=Metallic Phase with Long-Range Orientational Order and No Translational Symmetry |url=https://link.aps.org/doi/10.1103/PhysRevLett.53.1951 |journal=Physical Review Letters |language=en |volume=53 |issue=20 |pages=1951–1953 |bibcode=1984PhRvL..53.1951S |doi=10.1103/PhysRevLett.53.1951 |issn=0031-9007}}</ref> While there are similarities, quasicrystals are now considered to be a class of packing which is different from fivelings and the related icosahedral particles.<ref>{{Cite journal |date=2016-09-30 |title=NIST and the Nobel (September 30, 2016, Updated November 17, 2019) The Nobel Moment: Dan Shechtman |url=https://www.nist.gov/nist-and-nobel/dan-shechtman/nobel-moment-dan-shechtman |journal=NIST |language=en}}</ref><ref>{{Cite web |title=Quasicrystal - Online Dictionary of Crystallography |url=https://dictionary.iucr.org/Quasicrystal |access-date=2024-04-04 |website=dictionary.iucr.org}}</ref>

=== Heterogeneous catalysts ===
There are possible links to [[heterogeneous catalysis]], with the decahedral particles displaying different performance.<ref name=":44" /><ref name=":45" /><ref name=":46" /><ref name=":47" /> The first study by Avery and Sanders<ref name=":44">{{Cite journal |last1=Avery |first1=N |last2=Sanders |first2=J V |date=1970 |title=The structure of metallic particles in dispersed catalysts |url=https://linkinghub.elsevier.com/retrieve/pii/0021951770901715 |journal=Journal of Catalysis |language=en |volume=18 |issue=2 |pages=129–132 |doi=10.1016/0021-9517(70)90171-5}}</ref> did not find them in automobile catalysts. Later work by Marks and Howie found them in silver catalysts,<ref name=":45">{{Cite journal |last1=Marks |first1=L. D. |last2=Howie |first2=A. |date=1979 |title=Multiply-twinned particles in silver catalysts |url=https://www.nature.com/articles/282196a0 |journal=Nature |language=en |volume=282 |issue=5735 |pages=196–198 |doi=10.1038/282196a0 |bibcode=1979Natur.282..196M |issn=0028-0836}}</ref> and there have been other reports. It has been suggested that the strain at the surface can change reaction rates,<ref name=":46">{{Cite journal |last1=Walsh |first1=Michael J. |last2=Yoshida |first2=Kenta |last3=Kuwabara |first3=Akihide |last4=Pay |first4=Mungo L. |last5=Gai |first5=Pratibha L. |last6=Boyes |first6=Edward D. |date=2012 |title=On the Structural Origin of the Catalytic Properties of Inherently Strained Ultrasmall Decahedral Gold Nanoparticles |url=https://pubs.acs.org/doi/10.1021/nl300067q |journal=Nano Letters |language=en |volume=12 |issue=4 |pages=2027–2031 |doi=10.1021/nl300067q |pmid=22385208 |arxiv=1705.05763 |bibcode=2012NanoL..12.2027W |issn=1530-6984}}</ref> and since there is evidence that surface strain can change the adsorption of molecules and catalysis there is circumstantial support for this.<ref>{{Cite journal |last1=Grabow |first1=Lars |last2=Xu |first2=Ye |last3=Mavrikakis |first3=Manos |date=2006 |title=Lattice strain effects on CO oxidation on Pt(111) |url=http://xlink.rsc.org/?DOI=b606131a |journal=Physical Chemistry Chemical Physics |language=en |volume=8 |issue=29 |pages=3369–3374 |doi=10.1039/b606131a |pmid=16855712 |bibcode=2006PCCP....8.3369G |issn=1463-9076}}</ref><ref>{{Cite journal |last1=Liu |first1=Fuzhu |last2=Wu |first2=Chao |last3=Yang |first3=Guang |last4=Yang |first4=Shengchun |date=2015 |title=CO Oxidation over Strained Pt(100) Surface: A DFT Study |url=https://pubs.acs.org/doi/10.1021/acs.jpcc.5b04511 |journal=The Journal of Physical Chemistry C |language=en |volume=119 |issue=27 |pages=15500–15505 |doi=10.1021/acs.jpcc.5b04511 |issn=1932-7447}}</ref> {{As of|2024}}, there is some experimental evidence for different catalytic reactivity.<ref>{{Cite journal |last1=Wang |first1=Qiyu |last2=Cui |first2=Xiaoqiang |last3=Guan |first3=Weiming |last4=Zhang |first4=Lei |last5=Fan |first5=Xiaofeng |last6=Shi |first6=Zhan |last7=Zheng |first7=Weitao |date=2014 |title=Shape-dependent catalytic activity of oxygen reduction reaction (ORR) on silver nanodecahedra and nanocubes |url=https://linkinghub.elsevier.com/retrieve/pii/S0378775314010349 |journal=Journal of Power Sources |language=en |volume=269 |pages=152–157 |doi=10.1016/j.jpowsour.2014.06.160|bibcode=2014JPS...269..152W }}</ref><ref name=":47">{{Cite journal |last1=Zhou |first1=Yuheng |last2=Zhu |first2=Yihan |last3=Wang |first3=Zhi-Qiang |last4=Zou |first4=Shihui |last5=Ma |first5=Guicen |last6=Xia |first6=Ming |last7=Kong |first7=Xueqian |last8=Xiao |first8=Liping |last9=Gong |first9=Xue-Qing |last10=Fan |first10=Jie |date=2017 |title=Catalytic Activity Control via Crossover between Two Different Microstructures |url=https://pubs.acs.org/doi/10.1021/jacs.7b05476 |journal=Journal of the American Chemical Society |language=en |volume=139 |issue=39 |pages=13740–13748 |doi=10.1021/jacs.7b05476 |pmid=28885842 |bibcode=2017JAChS.13913740Z |hdl=10754/625450 |issn=0002-7863|hdl-access=free }}</ref><ref>{{Cite journal |last1=Choi |first1=Chungseok |last2=Cheng |first2=Tao |last3=Flores Espinosa |first3=Michelle |last4=Fei |first4=Huilong |last5=Duan |first5=Xiangfeng |last6=Goddard |first6=William A. |last7=Huang |first7=Yu |date=2019 |title=A Highly Active Star Decahedron Cu Nanocatalyst for Hydrocarbon Production at Low Overpotentials |url=https://onlinelibrary.wiley.com/doi/10.1002/adma.201805405 |journal=Advanced Materials |language=en |volume=31 |issue=6 |pages=e1805405 |doi=10.1002/adma.201805405 |pmid=30549121 |bibcode=2019AdM....3105405C |osti=1610774 |issn=0935-9648}}</ref>

=== Plasmonics ===
It is known that the response of the [[surface plasmon polariton]]s in nanoparticles depends upon their shape.<ref>{{Cite journal |last1=Eustis |first1=Susie |last2=El-Sayed |first2=Mostafa A. |date=2006 |title=Why gold nanoparticles are more precious than pretty gold: Noble metal surface plasmon resonance and its enhancement of the radiative and nonradiative properties of nanocrystals of different shapes |url=http://xlink.rsc.org/?DOI=B514191E |journal=Chem. Soc. Rev. |language=en |volume=35 |issue=3 |pages=209–217 |doi=10.1039/B514191E |pmid=16505915 |issn=0306-0012}}</ref> As a consequence decahedral particles have specific optical responses.<ref>{{Cite journal |last1=Rodríguez-Fernández |first1=Jessica |last2=Novo |first2=Carolina |last3=Myroshnychenko |first3=Viktor |last4=Funston |first4=Alison M. |last5=Sánchez-Iglesias |first5=Ana |last6=Pastoriza-Santos |first6=Isabel |last7=Pérez-Juste |first7=Jorge |last8=García de Abajo |first8=F. Javier |last9=Liz-Marzán |first9=Luis M. |last10=Mulvaney |first10=Paul |date=2009 |title=Spectroscopy, Imaging, and Modeling of Individual Gold Decahedra |url=https://pubs.acs.org/doi/10.1021/jp907646d |journal=The Journal of Physical Chemistry C |language=en |volume=113 |issue=43 |pages=18623–18631 |doi=10.1021/jp907646d |issn=1932-7447}}</ref><ref>{{Cite journal |last1=Pietrobon |first1=Brendan |last2=McEachran |first2=Matthew |last3=Kitaev |first3=Vladimir |date=2009 |title=Synthesis of Size-Controlled Faceted Pentagonal Silver Nanorods with Tunable Plasmonic Properties and Self-Assembly of These Nanorods |url=https://pubs.acs.org/doi/10.1021/nn800591y |journal=ACS Nano |language=en |volume=3 |issue=1 |pages=21–26 |doi=10.1021/nn800591y |pmid=19206244 |issn=1936-0851}}</ref> One suggested use is to improve light adsorption using their plasmonic properties by adding them to polymer [[solar cell]]s.<ref>{{Cite journal |last1=Jheng |first1=Jhih-Yuan |last2=Sah |first2=Pai-Tao |last3=Chang |first3=Wei-Che |last4=Chen |first4=Jhe-Han |last5=Chan |first5=Li-Hsin |date=2017 |title=Decahedral gold nanoparticles for enhancing performance of polymer solar cells |url=https://linkinghub.elsevier.com/retrieve/pii/S0143720816309445 |journal=Dyes and Pigments |language=en |volume=138 |pages=83–89 |doi=10.1016/j.dyepig.2016.11.027}}</ref>
[[File:Mechanical-5ling.tif|thumb|Five-fold twin at an Au tip after tensile failure.<ref name=":42" /> The scale bar is 2 nm.]]

=== Thin films and mechanical deformation ===
Most observations of fivelings have been for isolated particles. Similar structures can occur in thin films when particles merge to form a continuous coating, but do not recrystallize immediately.<ref>{{Cite journal |last=Marks |first=L.D. |date=1986 |title=Solid-like growth |url=https://linkinghub.elsevier.com/retrieve/pii/0040609086902907 |journal=Thin Solid Films |language=en |volume=136 |issue=2 |pages=309–315 |doi=10.1016/0040-6090(86)90290-7|bibcode=1986TSF...136..309M }}</ref><ref>{{Cite journal |last1=Bikmukhametov |first1=Ilias |last2=Tucker |first2=Garritt J. |last3=Thompson |first3=Gregory B. |date=2024 |title=Five-fold twin structures in sputter-deposited nickel alloy films |url=https://www.sciencedirect.com/science/article/pii/S1359646223005870 |journal=Scripta Materialia |volume=241 |pages=115866 |doi=10.1016/j.scriptamat.2023.115866 |issn=1359-6462}}</ref> They can also form during annealing of films,<ref name=":41">{{Cite journal |last1=Huang |first1=P. |last2=Dai |first2=G. Q. |last3=Wang |first3=F. |last4=Xu |first4=K. W. |last5=Li |first5=Y. H. |date=2009 |title=Fivefold annealing twin in nanocrystalline Cu |url=https://pubs.aip.org/apl/article/95/20/203101/321721/Fivefold-annealing-twin-in-nanocrystalline-Cu |journal=Applied Physics Letters |language=en |volume=95 |issue=20 |doi=10.1063/1.3263948 |bibcode=2009ApPhL..95t3101H |issn=0003-6951}}</ref><ref>{{Cite journal |last1=Parajuli |first1=Prakash |last2=Mendoza-Cruz |first2=Ruben |last3=Velazquez-Salazar |first3=J. Jesus |last4=Yacaman |first4=Miguel Jose |last5=Ponce |first5=Arturo |date=2019 |title=Fivefold annealing twin in nanocrystalline Au/Pd film |journal=Materials Letters |volume=244 |pages=88–91 |doi=10.1016/j.matlet.2019.02.060 |issn=0167-577X|doi-access=free |bibcode=2019MatL..244...88P }}</ref> which molecular dynamics simulations have indicated correlates to the motion of twin boundaries and a disclination,<ref>{{Cite journal |last1=Bringa |first1=E |last2=Farkas |first2=D |last3=Caro |first3=A |last4=Wang |first4=Y |last5=Mcnaney |first5=J |last6=Smith |first6=R |date=2008 |title=Fivefold twin formation during annealing of nanocrystalline Cu |url=https://linkinghub.elsevier.com/retrieve/pii/S1359646208006398 |journal=Scripta Materialia |language=en |volume=59 |issue=12 |pages=1267–1270 |doi=10.1016/j.scriptamat.2008.08.041|osti=966234 }}</ref> similar to the case of isolated nanoparticles described [[#DMove|earlier]]. There is experimental evidence in thin films for interactions between partial dislocations and disclinations,<ref>{{Cite journal |last1=Chen |first1=Yingbin |last2=Huang |first2=Qishan |last3=Zhao |first3=Shuchun |last4=Zhou |first4=Haofei |last5=Wang |first5=Jiangwei |date=2021 |title=Interactions between Dislocations and Penta-Twins in Metallic Nanocrystals |journal=Metals |language=en |volume=11 |issue=11 |pages=1775 |doi=10.3390/met11111775 |doi-access=free |issn=2075-4701}}</ref> as discussed in 1971 by de Wit.<ref name=":40" /> They can also be formed by mechanical deformation.<ref name=":42">{{Cite journal |last1=Wang |first1=Xiang |last2=Zheng |first2=Sixue |last3=Deng |first3=Chuang |last4=Weinberger |first4=Christopher R. |last5=Wang |first5=Guofeng |last6=Mao |first6=Scott X. |date=2023 |title=In Situ Atomic-Scale Observation of 5-Fold Twin Formation in Nanoscale Crystal under Mechanical Loading |journal=Nano Letters |language=en |volume=23 |issue=2 |pages=514–522 |doi=10.1021/acs.nanolett.2c03852 |issn=1530-6984 |pmc=10032584 |pmid=36633548|bibcode=2023NanoL..23..514W }}</ref> The formation of a local fiveling structure by annealing or deformation has been attributed to a combination of stress relief and twin motion,<ref name=":41" /><ref name=":42" /><ref>{{Cite journal |last1=Thomas |first1=Spencer L. |last2=King |first2=Alexander H. |last3=Srolovitz |first3=David J. |date=2016 |title=When twins collide: Twin junctions in nanocrystalline nickel |journal=Acta Materialia |language=en |volume=113 |pages=301–310 |doi=10.1016/j.actamat.2016.04.030|bibcode=2016AcMat.113..301T |doi-access=free }}</ref> which is different from the surface energy driven formation of isolated particles described [[#Energy versus size|above]].

== See also ==
*{{Annotated link|Chemical physics}}
*{{Annotated link|Cluster (chemistry)}}
*{{annotated link|Cluster (physics)}}
*{{Annotated link|Crystal habit}}
*{{Annotated link|Crystal twinning}}
*{{Annotated link|Disclination}}
*{{Annotated link|Icosahedral twins}}
*{{Annotated link|Nanocluster}}
*{{Annotated link|Nanomaterials}}
*{{Annotated link|Nanowire}}
*{{Annotated link|Nucleation}}
*{{Annotated link|Surface energy}}
*{{Annotated link|Surface stress}}
*{{Annotated link|Wulff construction}}

== Notes==
{{notelist|refs=

{{efn|name=crystal|In [[mineralogy]] millimeter sized objects are normally referred to as <em>crystals</em>. In other areas the terms are different. When a fiveling has only a very few atoms, for instance the smallest which is seven, it would be called a <em>cluster</em>. They are also sometimes called <em>nucleii</em> or <em>seeds</em>. In the size range 2-100 nm they are currently called <em>nanoparticles</em>, although earlier names are <em>small particles</em> and <em>fine particles</em>.}}
{{efn|name=facets|In the nanoparticle literature as well as physics and chemistry the term <em>facet</em> is common for flat external surfaces, which is how it is used herein. In the mineralogical literature the term <em>facet</em> is more commonly used for the external surfaces created on the surfaces of gemstones by cutting and polishing, and <em>surface faces</em> is used for native crystallographic surfaces such as {111}, which are also sometimes called <em>natural facets</em>.}}
{{efn|name=symmetry|Common usage is to connect point group names to the corresponding shapes in two dimensions, such as pentagonal with pentagon, and polyhedra in three dimensions such as decahedral for a [[decahedron]] (pentagonal bipyramid) and icosahedral for [[icosahedron]].}} 
}}

== References ==
{{reflist}}

== External links ==
*{{Cite web |title=Crystal creator code |url=https://www.on.msm.cam.ac.uk/code.html |access-date=2024-04-01 |website=www.on.msm.cam.ac.uk |language=en}} Code from the group of [[Emilie Ringe]] which calculates thermodynamic and kinetic shapes for decahedral particles and also does optical simulations, see also {{Cite journal |last1=Boukouvala |first1=Christina |last2=Ringe |first2=Emilie |date=2019-10-17 |title=Wulff-Based Approach to Modeling the Plasmonic Response of Single Crystal, Twinned, and Core–Shell Nanoparticles |journal=The Journal of Physical Chemistry C |language=en |volume=123 |issue=41 |pages=25501–25508 |doi=10.1021/acs.jpcc.9b07584 |issn=1932-7447 |pmc=6822593 |pmid=31681455}}.
*{{Cite web |title=WulffPack – a package for Wulff constructions |url=https://wulffpack.materialsmodeling.org/ |access-date=2024-04-01 |website=wulffpack.materialsmodeling.org |language=en}} Code from J M Rahm and P Erhart which calculates thermodynamic shapes, both continuum and atomistic, see also {{Cite journal |last1=Rahm |first1=J. |last2=Erhart |first2=Paul |date=2020 |title=WulffPack: A Python package for Wulff constructions |journal=Journal of Open Source Software |volume=5 |issue=45 |pages=1944 |doi=10.21105/joss.01944 |bibcode=2020JOSS....5.1944R |issn=2475-9066|doi-access=free }}.
*{{Cite web |title=Shape Software |url=https://www.shapesoftware.com/00_Website_Homepage/ |access-date=2024-05-09 |website=www.shapesoftware.com}} The code can be used to generate thermodynamic Wulff shapes including twinning.

[[Category:Chemical physics]]
[[Category:Condensed matter physics]]
[[Category:Crystallography]]
[[Category:Materials science]]
[[Category:Mineralogy]]
[[Category:Nanoparticles]]
[[Category:Physical chemistry]]
[[Category:Solid-state chemistry]]