11 PolymerCrystallization 11.1INTRODUCTION Low-molecular-weightmaterials,suchasmetals,typicallyexistascrystalsinthe solidstate.Thedrivingforcebehindtheformationofcrystals,whichare structureshavingalong-rangeperiodicorder,istheloweringinfreeenergy thataccompaniestheprocessofcrystallization.Thus,ifweweretoplottheGibbs freeenergyperunitvolume,G,ofamaterialinboththesolidcrystallineand moltenformsasafunctionoftemperature,wewouldgetaplotofthetypeshown inFigure11.1;thedecreaseinfreeenergywithincreasingtemperatureforboth phases is due to the relative increase in the temperature–entropy term. The point of intersection of the two curves is the equilibrium melting point T 0 M , whereas the vertical difference between them represents the free-energy change, DG v , between the two states at any temperature. Note that many materials (such as iron) exhibit polymorphism; that is, they exist in more than one crystalline form. In such a case, each form has its own G versus temperat ure curve. Long-chain molecules can also crystallize, and they do so for the same energetic reasons as short-chain molecules. However, for a polymer to be crystallizable, its chemical structure should be regular enough that the polymer molecule can arrange itself into a crystal lattice. Thus, isotactic and syndiotactic polypropylenes crystallize easily, but atactic polypropylene does not. For the same reason, the presence of bulky side groups (as in polystyrene) hinders crystallization, but the possibility of hydrogen-bonding (as in polyamides) 437 Copyright © 2003 Marcel Dekker, Inc. promotestheprocess.Figure11.2showsaschematicrepresentationofnylon66 [1].Aunitcellcontainsatotalofonechemicalrepeatunit,andthemoleculesare inthefullyextendedzigzagconformation.Thepolymerchaindirectionis generallylabeledthecaxisofthecell.Thecellitselfistriclinic,whichmeans thatmagnitudesofthethreeaxesandthethreeinteraxialanglesarealldifferent. AsketchoftheunitcellisshowninFigure11.3,inwhicharrowsareusedto designatehydrogenbonds.Whetheracrystallizablepolymeractuallycrystallizes ornot,though,dependsonthethermalhistoryofthesample.Formacromole- cules,polymermobilityexistsonlyabovetheglasstransitiontemperature,and becausetheenergeticsarefavorableonlybelowthemeltingpoint,crystallization cantakeplaceonlyinatemperaturerangebetweenT 0 m andT g .However, crystallizationisnotaninstantaneousprocess;ittakesplacebynucleationand growth,andthesestepstaketime.Iftherateofcoolingfromthemeltisrapid enough,acompletelyamorphouspolymercanresult.Thisisshownschematically inFigure11.4,whichisacontinuous-coolingtransformationcurve[2].Ifthe coolingrateissuchthatwecangofromT 0 m toT g withoutintersectingthecurve labeled‘‘crystallizationbeginsinquiescentmelts,’’nocrystallizationtakesplace. Therefore,itispossibletoobtaincompletelyamorphoussamplesofaslowly crystallizingpolymersuchaspolyethyleneterephthalate,butitisnotpossiblefor arapidlycrystallizingpolymersuchaspolyethylene.Inadditiontotemperature, theextentofcrystallizationalsodependsonfactorssuchastheappliedstress duringprocessing,whichtendstoalignpolymerchainsinthestressdirection. Thiscanaltertheenergeticsofphasechangeandcanleadtoaverysignificant enhancementoftherateofcrystallization.Thephenomenoncanbeunderstoodas ashifttotheleftinFigure11.4,fromthecurveindicatingtheonsetofquiescent crystallizationtothecurvelabeled‘‘crystallizationbeginsinstretchedmelts.’’ Becausepolymersarerarelycompletelycrystalline,theyarecalledsemicrystal- line. FIGURE11.1VariationwithtemperatureoftheGibbsfreeenergyperunitvolume. 438 Chapter 11 Copyright © 2003 Marcel Dekker, Inc. FIGURE 11.2 Schematic representation of nylon 66. (From Ref. 1.) FIGURE 11.3 Perspective drawing of a unit cell of nylon 66. The viewpoint is 11 A ˚ up, 10 A ˚ to the right, and 40 A ˚ back from the lower left corner of the cell. (From Ref. 1.) Polymer Crystalliza tion 439 Copyright © 2003 Marcel Dekker, Inc. Crystallizable polymers that dissolve in a solvent can also be made to crystallize from solution. When this is done using dilute solutions, single crystals can be obtained [3]. The crystals can have a large degree of perfection and are usually in the form of lamellae or platelets having thickness of the order of 100 A ˚ and lateral dimensions of the order of microns. The observed thickness depends on the temperature of crystallization. As shown in Figure 11.5, a single lamella is composed of chain-folded polymer molecules. Lamellae of different polymers have been observed in the form of hollow pyramids and hexagonal structures. When crystallized from quiescent melts, though, spherical structures called spherulites are observed. These spheres, which can grow to a few hundred microns in diameter, are made up of lamellae that arrange themselves along the FIGURE 11.4 Schematic illustration of the concept of a ‘‘continuous-cooling trans- formation curve’’ showing the anticipated effect of stress in shifting such curves. (From Ref. 2.) FIGURE 11.5 A chain-folded crystal lamella. 440 Chapter 11 Copyright © 2003 Marcel Dekker, Inc. radial direction in the sphere, as shown in Figure 11.6. The interlamellar regions as well as the region between spherulites are composed of amorphous or noncrystallizable fractions of the polymer. When crystallization takes place from a strained (deforming) solution or strained melt, the crystal shape can change to that of a shish kebab in the former case and to a row-nucleated crystal in the latter case; these are shown in Figure 11.7 [4]. The chain orientation in solution initially results in the formation of extended chain crystals, which give rise to the central core or ‘‘shish’’; the ‘‘kebabs,’’ which are lamellar, then grow radially outward from the shish. If the polymer sample is polydisperse, the higher- molecular-weight fraction crystallizes first, and this results in fractionation. From this discussion, it should be clear that a solid semicrystalline polymer is a two-phase structure consisting essentially of an amorphous phase with a FIGURE 11.6 Schematic diagram of a spherulite. Each ray is a lamella. FIGURE 11.7 Oriented morphologies appearing in polyethylene. (Reprinted from Gedde, U. W., Polymer Physics, Figure 7.38, copyright 1995, Chapman and Hall. With kind permission of Kluwer Academic.) Polymer Crystalliza tion 441 Copyright © 2003 Marcel Dekker, Inc. dispersed crystalline phase. Each phase is characterized by different values of a given physical property. The density of the crystals, for instance, is always greater than that of the amorphous polymer. Furthermore, process conditions determine the volume fraction of crystals, their shape, size, and size distribution, the orientation of polymer chains within the two phases, and how the crystalline regions are connected to the amorphous regions. Thus, whereas the properties of an amorphous polymer can be described as glassy or rubbery (depending on whether the temperature of measurement is below or above the glass transition temperature), the behavior of a semicrystalline polymer is much more compli- cated and is often anisotropic: If polymer chains are aligned in a particular direction, the material will be very strong in that direction but weak in a direction perpendicular to it. However, one of the two phases may be dominant in terms of influencing a particular overall property of the polymer. Thus, regarding mechan- ical properties, we find that increasing the spherulite size results in a decrease in the impact strength, an increase in the yield stress, and a reduction in the elongation to break in a tensile experiment while the Young’s modulus goes through a maximum [5]. The solubility of a molecule in a polymer and also its diffusivity, though, are determined by the amorphous phase. As a consequence, the permeability, which is a product of these two quantities, decreases as the extent of crystallinity increases. The breakdown of electrical insulation, on the other hand, depends on the properties of the interspherulitic region in the polymer [6]. Other factors that are influenced by the structure include brittleness, environmental degradation, thermal properties, melting point, and glass transition temperature. If the solid structure that is formed is a nonequilibrium one, it can change later if conditions (especially temperature) are such that equilibrium can be approached. Thus, a polymer sample whose chains have been frozen in an extended position can shrink when chain alignment is lost on heating to a temperature above the glass transition temperature. Although some of the influence of structure on properties can be rationa- lized by thinking of crystallites either as filler particles in an amorphous matrix or as permanent cross-links (as in vulcanized rubber), a proper understanding of structure development during processing is necessary to satisfy intellectual curiosity and to utilize crystallization knowledge for economic gain. As a result, we need to know what structure arises from a given set of processing conditions, how we can characterize (or measure) this structure, and how it affects a property of interest. It is, of course, much more difficult to reverse the process of thinking and inquire how we might obtain a particular structure in order to get specified values of properties of interest. This is the realm of engineered material properties and the subject of research of many industrial research laboratories. Before tackling greater problems, though, we must first get acquainted with some rather fundamental concepts. 442 Chapter 11 Copyright © 2003 Marcel Dekker, Inc. 11.2ENERGETICSOFPHASECHANGE Ifthetemperatureofaliquidisloweredtobelowthemeltingpoint,materialtends tosolidify.Asmentionedpreviously,theprocessisneithersuddennorinstanta- neous.Indeed,itproceedsrelativelyslowlyandonasmallscaleifthetemperature isonlyslightlybelowthemeltingpoint,anditinvolvestwodistinctsteps. Initially,nucleiofthenewphasemustbeformed,andtheeasewithwhichthis happensdependsontheextentofsupercooling.Thisstepisfollowedbygrowth ofthenuclei,aprocedurethatinvolvesdiffusionofmaterialtothephase boundary.Thecombinedprocessofnucleationandgrowthisthesameregardless ofwhetherthecrystallizationbehaviorbeingobservedisthatofmoltenmetalsor moltenpolymers. 11.2.1HomogeneousNucleation Tounderstandthethermodynamicsofnucleation,letusfirstconsiderhomo- geneousnucleation,alsocalledsporadicnucleation,fromanisothermal,quies- centmeltwhosetemperatureTiskeptbelowthemeltingpointT 0 m .Here, ‘‘homogeneous’’referstotheappearanceofthenewsolidphaseinthemiddle oftheoldliquidphase. BasedonFigure11.1,wewouldexpectnucleationtobeaccompaniedbya reduction in the Gibbs free energy equal to DG v per unit volume. However, the system free energy is not reduced by the full amount of DG v . This is because surface energy equal to g per unit area has to be expended in creating the surface that bounds the nuclei. Thus, if the typical nucleus is a sphere of radius r, the net change in the free energy due to the formation of this particle is DG ¼ 4pr 2 g þ 4 3 pr 3 DG v ð11:2:1Þ where DG v is a negative number. For a small sphere, the surface area-to-volume ratio can be fairly large; therefore, DG initially increases with increasing r and goes through a maximum before becoming negative. This maximum (positive) value DG* represents an energy barrier and must be overcome by the thermal motion of the molecules before a stable nucleus can be formed. If we set the derivative of DG with respect to r equal to zero, then using Eq. (11.2.1) we find that r*, the value of r corresponding to DG*, is r* ¼À 2g DG v ð11:2:2Þ with the following result: DG* ¼ 16pg 3 3ðDG v Þ 2 ð11:2:3Þ Polymer Crystalliza tion 443 Copyright © 2003 Marcel Dekker, Inc. Because the magnitude of DG v increases as the temperature is lowered, both r* and DG* decrease with decreasing temperature. This variation can be made explicit by noting that, at the melting point, DG v ¼ DH v À T 0 m DS v ¼ 0 ð11:2:4Þ where H and S are respectively the enthalpy and entropy per unit volume. Consequently, we have the following: DS v ¼ DH v T 0 m ð11:2:5Þ Now, DS v and DH v depend only weakly on temperature, so that DG v ¼ DH v 1 À T T 0 m ð11:2:6Þ which, when introduced into Eqs. (11.2.2) and (11.2.3), gives r* ¼À 2gT 0 m DH v DT ð11:2:7Þ DG* ¼ 16pg 3 ðT 0 m Þ 2 3DH 2 v DT 2 ð11:2:8Þ where D T equals the amount of subcooling (T 0 m À T) and DH v is physically the latent heat of crystallization per unit volume and is a negative quantity. Clearly, lower temperatures favor the process of nucleation, as DG* decreases rapidly with decreasing temperature. Example 11.1: For a polyolefin it is found that DH v ¼À3  10 9 ergs=cm 3 and g ¼ 90 ergs=cm 2 . If the equilibrium melting point is 145 C, how do the radius r* of a critical-sized nucleus and the associated energy change DG* depend on the extent of subcooling, DT ? Solution: Using Eq. (11.2.7), we find that r*DT ¼ 2508 where r* is measured in angstroms. Also, with the help of Eq. (11.2.8), we have DG*ðDTÞ 2 ¼ 2:37  10 À7 ergs K 2 11.2.2 Heterogeneous Nucleation Although the treatment of the previous section can be extended to nonspherical nuclei, we find that this is not needed in practice because the contribution of 444 Chapter 11 Copyright © 2003 Marcel Dekker, Inc. homogeneous nucleation to overall crystal growth is small compared to that of heterogeneous or predetermined nucleation, unless the temperature is signifi- cantly below the melting point. In the case of heterogeneous nucleation, crystal growth takes place on a pre-existing surface, which might be a dust particle, an impurity, part of the surface of the container, or an incompletely melted crystal. If we consider heterogeneous nucleation to take place on the surface of a pre- existing lamella, as shown in Figure 11.8 [7], we discover that if the crystal volume increases by an amount nabl, where abl is the volume of a single strand, the surface area increases by only 2bðl þ naÞ. Had we considered primary nucleation, the surface area would have gone up by 2bðl þ naÞþ2nal. Note that due to chain folding, g e , the surface energy associated with the chain ends can be expected to be large compared to g, the surface energy of the lateral surface. In view of the foregoing, the free-energy change due to the deposition of n polymer strands is DG ¼ 2blg þ 2bnag e þ nablDG v ð11:2:9Þ and the free-energy change involved in laying down the (n þ 1)st strand is obtained from Eq. (11.2.9) as DGðn þ 1ÞÀDGðnÞ¼2abg e þ ablDG v ð11:2:10Þ Clearly, for this process to be energetically favorable, the right-hand side of Eq. (11.2.10) has to be negative. This condition requires that [8] l > À 2g e DG v ð11:2:11Þ FIGURE 11.8 Crystal growth on a pre-existing surface. Polymer Crystalliza tion 445 Copyright © 2003 Marcel Dekker, Inc. which, in view of Eq. (11.2.6), implies that l > À 2g e T 0 m ðDH v DTÞ ð11:2:12Þ Because the argument leading up to Eq. (11.2.12) is valid for any value of n,it must also hold for the beginning of the process when n equals unity. Thus, for heterogeneous nucleation, the critical-sized nucleus occurs at n ¼ 1, with the result that DG* ¼ 2bl*g þ abl*DG v þ 2abg e ð11:2:13Þ with l* ¼À 2g e T 0 m DH v DT ð11:2:14Þ and we find that, just as with r*, l* depends inversely on DT. This is found to be true experimentally. Also, to a good approximation, DG* / 1 DT ð11:2:15Þ Example 11.2: Use the data given in Example 11.1 to confir m the validity of Eq. (11.2.15). Determine DG* for DT ¼ 10 K and compare this value with that in Example 11.1. Assume that g e % g. Solution: As g e % g, l* ¼ r* ¼ 250:8A ˚ . Because a cannot be more than a few angstroms, a ( l* and we can neglect the last term on the right-hand side of Eq. (11.2.13) in comparison with the first term. The second term in Eq. (11.2.13) can also be neglected, provided that the following holds: aDG v ( 2g or aDH v DT 2gT 0 m ( 1 Introducing numbers, aDH v DT=2gT 0 m ¼ að3  10 9  10Þ=ð2  90  418Þ¼ 3:98  10 5 a, which is much less than unity because a < 10 À7 cm. Consequently, DG* ¼ 2bl*g ¼ 4bgg e T 0 m =DH v DT / 1=DT. For homogeneous nucleation, DG* is equal to 2:37  10 À9 ergs. For heterogeneous nucleation, DG* ¼ 4:51b  10 À4 ergs. Because b < 10 À7 cm, the energy barrier for heterogeneous nucleation is much smaller than that for homogeneous nucleation when the temperature is close to the equilibrium melting point. 446 Chapter 11 Copyright © 2003 Marcel Dekker, Inc. [...]... conductivity of typical polymers that is responsible for the poor ‘‘melt quality’’ that is often observed: The polymer that leaves the extruder can consist of islands of relatively cold, unmelted polymer floating in very hot molten liquid 11.7 INFLUENCE OF POLYMER CHAIN EXTENSION AND ORIENTATION Early work on polymer crystallization dealt exclusively with isothermal crystallization in stress-free, unoriented polymers... processing of blends of condensation polymers, such as two polyamides or two polyesters, is the occurrence of interchange reactions [31] The result of these transamidation or transesterification reactions is the formation of a block copolymer Initially, a diblock copolymer is produced, but, with increasing processing time, this gives way to blocks of progressively smaller size; ultimately, a random copolymer... the molar volumes of the repeating units, w12 is the polymer polymer interaction parameter, and f2 is the volume fraction (In a volume fraction, the amorphous polymer is denoted by subscript 1 and the crystalline polymer by 0 subscript 2.) Note that for Tm to be smaller than Tm , w12 has to be negative This is consistent with the remark following Eq (9.6.3) of Chapter 9 that polymer polymer miscibility... semicrystalline polymer This has a profound effect on the crystallization behavior of the semicrystalline polymer [33] In particular, there is a reduction in the melting point of the crystals and a change in the glass transition temperature; the Tg of the resulting random copolymer can be estimated using Eq (9.6.1) of Chapter 9 that was earlier shown to be valid for miscible polymer blends There is also a... semicrystalline polymer or (2) forming oriented crystals either by employing an oriented melt or by stretching the glassy polymer before annealing In such cases, we also need to know the average orientation of polymer molecules relative to some axis in each phase This information is necessary for computing a particular average property of the polymer The orientation, relative to a specified direction, of the polymer. .. Fraction Crystallinity The simplest method of determining the mass fraction crystallinity X of an unfilled, semicrystalline homopolymer is to measure the density r of a representative sample If the material is free of voids and impurities, the total volume V of unit mass of polymer is given by V ¼ X 1ÀX þ rc ra ð11:9:2Þ where rc and ra are the known densities of the crystalline and amorphous phases, respectively... molding (see Chapter 15 for a description of the process) Because some of the very largevolume polymers such as polyethylene, polypropylene, and various nylons are injection molded, the question of the microstructure of semicrystalline polymers has received a considerable amount of attention [12] This has led to the formulation of empirical expressions based on the theory originally developed by Avrami... synthesis of copolymers is often simpler and more economical than making the copolymers in a chemical reactor, and ‘‘reactive extrusion’’ is a major industry today [32] If one homopolymer is Copyright © 2003 Marcel Dekker, Inc Polymer Crystallization 459 semicrystalline and the other amorphous, noncrystallizable sequences will be built in between the crystallizable sequences of the semicrystalline polymer. .. treatment of the temperature dependence of G Copyright © 2003 Marcel Dekker, Inc 454 Chapter 11 F IGURE 11 .12 Avrami plot of the data shown in Figure 11.11 Plot of logfÀ ln½1 À Xc ðtÞ=Xc ð1Þg versus time for isothermal crystallization at 315 C (u), 312 C (s), 308 C (n), 164 C (j), and 160 C (d) (From Ref 21.) Reprinted from Polymer, vol 27, Cebe, P., and S D Hong: ‘‘Crystallization Behaviour of Poly(ether-ether-ketone),’’... permeability, and all this happens without loss of any other property of interest Not surprisingly, nanocomposites are being researched for a wide variety of applications, including the original automotive applications In terms of the crystallization behavior of polymers containing nanofillers, it has been found that the presence of silicate layers enhances the rate of isothermal crystallization [35]; this . Inc. whichisknownastheAvramiequation.Allofthetemperaturedependenceis embodiedintherateconstantk,whereastheAvramiexponent,n,isusually consideredtobethesumpþq,withpbeing0or1dependingonpredetermined orsporadicnucleationandqbeing1,2,or3dependingonthedimensionalityof crystalgrowth.Thus,nwouldequal3forthegrowthofdisklikecrystalsby homogeneousnucleationbutwouldonlybe2ifthenucleationwerehetero- geneous. Formostpolymers,crystallinityisnevercompleteandEq.(11.4.10)is modifiedbydefininganeffectivefractionoftransformedmaterialX=X 1 ,where X 1 isthemassfractioncrystallizedattheendofthetransformation.Theresultof thismodificationisgivenasfollows[20]: 1À X X 1 ¼expÀ k X 1 t n ð11:4:11Þ or ln1À X X 1 ¼Àkt n ð11:4 :12 whereX 1 ontheright-handsideofEq.(11.4 .12) hasbeenabsorbedintothe constantk. Figure11.11showscrystallizationdataforthedegreeofcrystallinityXasa functionoftime,atseveralconstanttemperatures,forpoly(ether-ether-ketone) (PEEK).Thispolymerhasaglasstransitiontemperatureof145 Candamelting pointof340 C,propertiesthatmakeitacandidateforhigh-performance thermoplasticcompositematrixapplications.InFigure11.11,thedegreeof crystallinityisdeterminedastheratiobetweentheheatevolvedduringisothermal crystallizationinadifferentialscanningcalorimeterandthelatentheatof crystallizationofaperfectcrystal.AnexaminationofFigure11.11revealsthat acertaininductiontimeisneededbeforecrystallizationcommencesandthatX 1 theultimatecrystallinityatverylongtimesdependsonthetemperatureof crystallization. WhenthedataofFigure11.11areplottedonlogarithmiccoordinates,as suggestedbyEq.(11.4 .12) ,asetofparallellinesisobtained,showninFigure 11 .12. Theslopeofeachofthelinesisapproximately3,suggestinghetero- geneousnucleationandthree-dimensionalspheruliticgrowth.Note,though,that inthelatterstagesofcrystallization,growthslowsandtheAvramiexpressionis notobeyed.Thisphaseofcrystallizationiscalledsecondarycrystallization,and itischaracterizedbythethickeningofcrystallamellaeandanincreaseincrystal perfectionratherthananincreaseinspheruliteradius. Example11.4:Itisoftenfound(seeFig.11.13)thatprimaryisothermal crystallization. temperature of crystallization. As shown in Figure 11.5, a single lamella is composed of chain-folded polymer molecules. Lamellae of different polymers have been observed in the form of hollow. might obtain a particular structure in order to get specified values of properties of interest. This is the realm of engineered material properties and the subject of research of many industrial