Crystal packing PDF
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21 Crystal Packing by Angelo Gavezzotti and Howard Flack This electronic edition may be freely copied and redistributed for educational or research purposes only. Itmaynotbesoldforprofitnorincorporatedinanyproductsoldforprofitwithout theexpresspermissionofTheExecutiveSecretary,InternationalUnionofCrystal- lography,2AbbeySquare,ChesterCH12HU,UK. Copyrightinthiselectronicedition(cid:13)c2005InternationalUnionofCrystallography. http://www.iucr.org/iucr-top/comm/cteach/pamphlets/21/21.html InternationalUnionofCrystallography CommissiononCrystallographicTeaching CRYSTAL PACKING AngeloGavezzottia andHowardFlackb a DipartimentodiChimicaStrutturaleeStereochimicaInorganica, Universita` diMilano,Milan,Italy,andb Laboratoirede Cristallographie,Universite´ deGene`ve,Geneva,Switzerland 1 Introduction Weallknowbyeverydayexperiencethatmatterhasmanydifferentstatesofag- gregation. Chemistsalsoknowthatmatterismadeofatoms,ionsandmolecules, andthatthemacroscopicpropertiesofanyobjectdependonthesize, shapeand energiesofthesemicroscopicconstituents. Onemoleofgaseoussubstanceoccupiesabout24litresatroomtemperature, while the volume of the same amount of substance in the liquid or solid state is afewtenstoafewhundredmillilitres. Itfollowsthatthemolecules1 aremuch, muchclosertoeachotherinaliquidandasolidthaninagas.Aneasycalculation shows that in condensed phases the average volume per molecule is about one andahalftimesthevolumeofthemoleculeitself. Moleculesaretightlypacked inspace,andthereforethecompressibilityofcondensedmediaisverysmall. You cansitonarocksimplybecauseitsatomsandmoleculesaresoclosetoeachother thattheycannotgivewayunderexternalpressure. Agaswilldiffuseveryquicklyoutofanopenbottle,whileasolidcanusually be left in the open air almost indefinitely without apparent change in size and shape(thereareexceptions,likemothballs). Besidesrepellingeachotheratshort distances,moleculesinasolidarereluctanttoleavetheirneighbours;thismeans that some sort of attraction is holding them together. Temperature has a much moredramaticeffectonallthisthanpressure: ordinaryliquidsboilwhenheated mildly,andevensolidrockmeltsandvaporizesinvolcanicdepths. 1Fromnowon,thetermmoleculedenotesamoleculeproper,oranyotherchemicalentityalso recognizableinthegasphase(aheliumatom, anNa+ orSO24− ion, anFe2(CO)9 complex). In general,itcanbesaidthatamoleculeisadistinguishableentitywhentheforcesactingwithinitare muchstrongerthantheforcesactingonitinthecrystal. Difficultiesarisewithinfinitestringsor layers; diamondandNaClcrystalsareexamplesofthree-dimensionallyinfinitesystemswherethe termmoleculeismeaningless. Also, wheneverorganiccompoundsarementionedinthetext, one shouldreadorganicandorganometalliccompounds. 1 Throughsimplereasoningonelementaryevidence,weareledtothefollowing conclusions: upon cooling or with increasing pressure, molecules stick together and form liquid and solid bodies, in which the distance between them is of the same order of magnitude as the molecular dimensions; and an increasing repul- sionarisesiftheyareforcedintoclosercontact. Thereverseoccursuponheating orloweringtheexternalpressure. While a layman may be more than satisfied at this point, a scientist must ask him- or herself at least two further questions: (1) What is the nature and magnitudeoftheforcesholdingmoleculestogether? (2)Whatisthegeometrical arrangement of molecules at close contact? Restricting the scope, as we do in this pamphlet, to crystalline solids, these questions define the subject of crystal packing.Sincecrystalsareendowedwiththebeautifulgiftoforderandsymmetry, thespatialpart(2)isnottrivial. Packingforcesandcrystalsymmetrydetermine thechemicalandphysicalpropertiesofcrystallinematerials. 2 Thermodynamics and kinetics Now put yourself in the place of a molecule within a pure and perfect crystal, beingheatedbyanexternalsource. Atsomesharplydefinedtemperature,abell rings, you must leave your neighbours, and the complicated architecture of the crystalcollapsestothatofaliquid. Textbookthermodynamicssaysthatmelting occurs because the entropy gain in your system by spatial randomization of the molecules has overcome the enthalpy loss due to breaking the crystal packing forces: T[S(liquid)–S(solid)]>H(liquid)–H(solid) G(liquid)<G(solid) This rule suffers no exceptions when the temperature is rising. By the same token,oncoolingthemelt,attheverysametemperaturethebellshouldringagain, andmoleculesshouldclickbackintotheverysamecrystallineform. Theentropy decreaseduetotheorderingofmoleculeswithinthesystemisovercompensated bythethermalrandomizationofthesurroundings, duetothereleaseoftheheat offusion;theentropyoftheuniverseincreases. Butliquidsthatbehaveinthiswayoncoolingaretheexceptionratherthanthe rule; in spite of the second principle of thermodynamics, crystallization usually occurs at lower temperatures (supercooling). This can only mean that a crystal ismoreeasilydestroyedthanitisformed. Similarly,itisusuallymucheasierto dissolveaperfectcrystalinasolventthantogrowagainagoodcrystalfromthe resultingsolution.Thenucleationandgrowthofacrystalareunderkinetic,rather thanthermodynamic,control. 2 3 Forces Amoleculeconsistsofacollectionofpositivelychargedatomicnucleisurrounded by an electron cloud. Even if the molecule has no net charge, such an object can hardly be considered as electrically neutral. Its electrostatic potential is a superpositionofthefieldsofallnucleiandelectrons. Anapproachingchargecan alter, by its own electrostatic field, the electron distribution in a molecule; this phenomenoniscalledpolarization. Theattractiveforcesholdingmoleculestogetherareaconsequenceofmolec- ularelectrostaticpotentials. Forpurelyioniccrystals,onecanjustuseCoulomb’s lawwithintegercharges; fororganicmolecules,ittakesamorecomplicatedex- pression, involving an integration over continuous electron densities. Alterna- tively,thechargedistributioncanberepresentedbyaseriesexpansionusingmul- tipoles, and the interaction energy can be calculated as a function of multipole moments. Different atoms have different electronegativities. Larger charge separations within the molecule – in the jargon of the trade, more polar molecules – build up stronger intermolecular forces. Ionic crystals are very hard and stable, while naphthalene or camphor (two common ingredients of mothballs) sublime rather easily. Thesenon-polarhydrocarbonmoleculesmustrelyonmutualpolarization toproduceattraction;theresultingforcesarefeeble,andarecalleddispersionor vanderWaals’forces; theyareusuallydescribedbyempiricalformulae. Inthis way,evenargonmanagestoformasolidatverylowtemperature. Ubiquitous in crystals is the hydrogen bond, a polar interaction which is the mosteffectivemeansofrecognitionandattractionbetweenmolecules; soeffec- tive,thatmoleculeswithdonorandacceptorgroupsformhydrogenbondswithout exception.Thereisnocase(atleast,totheauthors’knowledge)whereamolecule thatcanformhydrogenbondsdoesnotdosointhecrystal. Therepulsionatshortintermoleculardistancearisesfromaquantummechan- icaleffect. AccordingtoPauli’sprinciple,electronswiththesamequantumnum- bers,nomatterifbelongingtodifferentmolecules,cannotoccupythesameregion ofspace. Thus,Pauli‘forces’–althoughtheyarenotforcesinthesenseofNew- tonian mechanics – steer electrons to mutual avoidance, and the total energy of theelectronsystemrisesifpairedelectronsarepulledtogether. Table1collectsthesimplepotentialsmentionedsofar.Directbutnon-specific measures of the strength of crystal forces are the melting temperature and the sublimationenthalpy. 4 Crystal symmetry Intermolecular attraction brings molecules together, but there is a priori no im- plication of order and symmetry. Glasses, in which molecules are oriented at 3 Table1: Formulaeforpotentialenergiesincrystals Electrostatic i.e. ions or point charges; q , q are the charges and R is the distance i j ij betweenthetwo: X E = (q q )/R . i j ij i,j Electrostatic(moleculesAandBwithelectrondistributionsσ andσ ): A B ZZ E = σ (r )σ (r )|r −r |−1dr dr . A 1 B 2 2 1 1 2 Dispersion-repulsion(A,B,C,D,m,...,Qareempiricalparameters;R isthedistance ij betweentwosites–usually,atomicnuclei–ondifferentmolecules): X E = Aexp(−BR )−CR−6+DR−m+···+QR−1. ij ij ij ij i,j Hydrogenbond:empiricalpotentialsinvolvinglocalcharges,localdipoles,etc.(thereisa varietyofapproachesintheliterature). random, are sometimes as stable as crystals, in which molecules are arranged in an ordered fashion. The ordering of irregularly shaped, electrically charged molecules does however imply anisotropy; for mechanical properties, it results inpreferentialcleavageplanes, whiletheconsequencesofoptical, electricaland magneticanisotropyleadtoavarietyoftechnologicalapplicationsofcrystalline materials. Butwhatisthelinkbetweenorder,symmetryandcrystalstability? Crystalsymmetry2 hastwofacets. Ononeside,inamilestonemathematical development, it was demonstrated that the possible arrangements of symmetry operations(inversionthroughapoint,rotation,mirrorreflection,translation,etc.) give rise to no less and no more than 230 independent three-dimensional space groups. After the advent of X-ray crystallography, space-group symmetry was determinedfromthesystematicabsencesindiffractionpatternsandusedtohelp inthecalculationofstructurefactorsandelectron-densitysyntheses. The other side of crystal symmetry has to do with the crystal structure, as resulting from mutual recognition of molecules to form a stable solid. This is a fascinating and essentially chemical subject, which requires an evaluation and a comparisonoftheattractiveforcesatworkinthecrystal. Space-groupsymmetry is needed here to construct a geometrical model of the crystal packing, and it comesintoplayinjudgingrelativestabilities. It should be clear that the necessary arrangements of symmetry operations 2The term crystal symmetry refers to microscopic relationships between molecules or parts of molecules,andnottomacroscopicmorphology. 4 Table2: Space-groupfrequenciesfrom[1]forasampleoforganiccrystals Rank Group No. of Moleculesin Point-group crystals generalposition symmetry 1 P2 /c 9056 8032(89%) 1 1 2 P2 2 2 4415 4415(100%) 1 1 1 1 3 P1 3285 2779(85%) 1 4 P2 2477 2477(100%) 1 1 5 C2/c 1371 802(58%) 2,1 6 Pbca 1180 1064(90%) 1 7 Pna2 445 445(100%) 1 1 8 P1 370 370(100%) 1 9 C2 275 225(82%) 2 10 Pnma 266 33(12%) m,1 12 Pbcn 205 94(46%) 1,2 14 P2 /m 127 40(31%) 1,m 1 16 P2 2 2 92 46(50%) 2 1 1 17 Fdd2 88 51(58%) 2 in space bear no immediate relationship to crystal chemistry. The fact that 230 spacegroupsexistdoesnotmeanthatmoleculescanfreelychooseamongthem whenpackinginacrystal. Farfromit, thereareratherstrictpackingconditions that must be met, and this can be accomplished only by a limited number of arrangements of very few symmetry operations; for organic compounds, these are inversion through a point (1), the twofold screw rotation (2) and the glide reflection(g). Somespacegroupsaremathematicallylegitimate,butchemically impossible, and the crystal structures of organic compounds so far determined belongtoaratherrestrictednumberofspacegroups[1,2,3](Table2). Whenchargeisevenlydistributedinamolecule,andthereisnopossibilityof forminghydrogenbonds, no specialanchoringpointsexist. Every region ofthe molecule has nearly the same potential for intermolecular attraction, and hence it is reasonable to expect that each molecule be surrounded by as many neigh- boursaspossible,formingasmanycontactsaspossible. Emptyspaceisawaste, and molecules will try to interlock and to find good space-filling arrangements. Thisclose-packingideaappearedveryearlyinitsprimitiveform[4],butwascon- sciouslyputforwardbyKitaigorodski[5]. Order and symmetry now come to the fore, since for an array of identical objects a periodic, ordered and symmetrical structure is a necessary (although notsufficient)conditionforanefficientclosepacking. Whenspecialinteractions (likehydrogenbonds)arepresent,theclose-packingrequirementmaybealittle 5 Figure1: Amoleculewithoutstronglypolarsitesorhydrogen-bondingcapabil- itychoosestoclose-packinthecrystal,bumpintohollows,inordertomaximize dispersiveinteractions.Whenstrongforcesarepresent,theclose-packingrequire- mentmaybelesscompelling(waterisanextremeandalmostuniqueexample). less stringent (Figure 1), but it turns out that all stable crystals have a packing coefficient3between0.65and0.80. 5 Symmetry operations Inacrystal,somesymmetryoperationscanbeclassifiedasintramolecular,mean- ing that they relate different parts of the same molecule and thus belong to the molecular point-group symmetry. The other symmetry operations, which act as true crystal-packing operators, may be called intermolecular, and these are the ones which relate different molecules in the crystal. This classification implies thatmoleculesbedistinguishableincrystals. The simplest intuitive way of viewing a symmetry operation is that it repro- ducesinspaceone,ormoreifappliedrepetitively,congruentorenantiomorphic 3Thepackingcoefficientistheratioofvolumeoccupiedbythemoleculesinthecelltothevolume ofthecell.Molecularvolumescanbecalculatedinanumberofways;thesimplestonesaredescribed byKitaigorodski[5],andothersbyGavezzotti[7]. 6 Figure2:Sketchesoftheeffectofsymmetryoperations.Top:inversionthrougha point. Below,left:twofoldscrewrotation;belowright,glidereflection. Thelatter twooperationsgiverisetostringsintheydirection. copiesofagivenobject,accordingtoawell-definedconvention(Figure2). The spatialrelationshipbetweentheparentandthereproducedmoleculesisstrict,so amoment’sponderingwillconvincethereaderthatsomeoperatorsaremoreef- fectivethanotherstowardsclosepacking. Forobjectsofirregularshape, mirror reflectionandtwofoldrotationproducebump-to-bumpconfrontation,whileinver- sion through a point, screw rotation and glide reflection favour bump-to-hollow, moreclose-packedarrangements(Figure3). Onemustnotforgetthatpuretrans- lation(t)isalwayspresentinacrystal. Exceptwheninfinitestringsorlayersare present,itisanintrinsicallyintermolecularoperator,whoseroleinclose-packing isprobablyintermediate(Figure4);spacegroupP1istheeighthmostpopulated onefororganicsubstances. The clearest proof of the leading role of 1, 2 and g in close packing comes 1 fromastatisticalanalysisofthespace-groupfrequenciesoforganiccompounds, care being taken to distinguish between inter- and intramolecular symmetry op- 7 Figure3: Amirrorreflection(mirrorplaneperpendiculartothepage,tracealong thesolidline)cannotproduceclose-packing. Translationalongsomedirectionis requiredtoallowinterlockingofmolecularshapes. Figure4: Atwo-dimensionalpatternobtainedbypuretranslation: notsobadfor interlocking and close packing. For a complete set of two-dimensional space- fillingdrawingsinallthe17planegroups,see[6]. 8
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