10. Nucleation and Crystallization

10. NUCLEATION AND CRYSTALLIZATION

By Gaylon S. Ross and Lois J. Frolen 10.1. Introduction 10.1.1. Aims and Objectives

It is well known that the physical, mechanical, and chemical properties of crystalline polymers depend on the morphology of the crystals formed as well as the degree of crystallinity of the sample. Although the study of the morphology of polymer crystals is both interesting and complex, this subject is beyond the scope of this part. Khoury and Passaglia’ have recently published a chapter on the morphology of crystalline synthetic polymers that is written for the nonspecialist and covers many details concerning morphological studies from the mid-1950s to the present. For more details concerning this fascinating subject the reader is referred to other excellent book^^-^ and review^.^-'^ This chapter is written for the nonspecialist. It includes a survey of some of the better experimental methods that are available for the study of polymer nucleation and crystallization. We are primarily concerned with the description and evaluation of the experimental methods rather than surveying the results obtained for individual polymers by different techniques. F. Khoury and E. Passaglia, Treatise Solid State Chem. 3, 335 (1976). H. Geil, “Polymer Single Crystals.” Wiley (Interscience), New York, 1963. B. Wunderlich, “Macromolecular Physics,” Vol. 1 . Academic Press, New York, 1973. * B. Wunderlich, “Macromolecular Physics,” Vol. 2. Academic Press, New York, 1976. A. Keller, in “Growth and Perfection of Crystals” (R. H. Doremus, B. W. Roberts, and D. Turnbull, eds.), p. 499. Wiley, New York, 1958. A. Keller, Rep. Prog. Phys. 31, Part 2, 623 (1968). ’ A. Keller, Kolloid-Z. & Z . Poly. 231, 386 (1969). A. Keller, Phys. Chem., Ser. One 8, 105 (1972). H . D. Keith, in “Physics and Chemistry of the Organic Solid State” (D. Fox, M. M. Labes, and A. Weissberger, eds.), p. 561. Wiley (Interscience), New York, 1963. lo H. D. Keith, Kolloid-Z. & Z. Polym., 231, 421 (1969). l1 P. Ingram and A. Peterlin, Encycl. Polym. Sci. Techno/.. 9, 204 (1968). l2 D. V. Rees and D. C. Bassett, J . Muter. Sci. 6, 1021 (1971). Is R. A. Fava, J. Polym. Sci., Part D 5 , l(1971).

* P.

339 METHODS O F EXPERIMENTAL PHYSICS, VOL. 16B

Contribution of the National Bureau of Standards. ISBN 0-12-415951-2

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Although we emphasize the study of crystallization in bulk samples, we discuss crystallization from solution to some extent. Information concerning the underlying theory will be provided along with the method for analyzing the results of each type of experiment.

10.2. General Background on Semicrystalline Polymers 10.2.1. Requirements for Crystallization

A polymer is produced synthetically by the systematic adding or linking of small low-molecular-weight units, monomers, producing a distribution of differing molecular-weight chains. The number of monomer units in each chain is referred to as the degree of polymerization. Such materials will crystallize if the resulting structures are such as to permit the chain backbones to be packed in an ordered parallel configuration. This requirement can only be met if nearly complete chemical and stereo regularity exists. If, in the polymerization, all the monomer units attach themselves in an orderly head-to-tail fashion, chemical regularity will exist. Occasionally a head-to-head (or tail-to-tail) attachment will occur that will present a flaw in the repeat regularity of the polymer backbone configuration. Other chemical imperfections can produce branching or side chains in the otherwise linear chain. When the monomer has asymmetrical side groups, a mere head-to-tail addition is not sufficient to produce a polymer chain capable of crystallizing. It must also be stereo regular. The side groups must be either (1) isotactic, i.e., all on the same side of the backbone, or (2) syndiotactic, i.e., an alternation, left-right-left along the backbone. If such side groups are randomly placed left or right along the polymer backbone, crystallization does not occur. For the purposes of this discussion, we only consider crystallizable polymers, namely, those possessing linear chains with few branch points. When the asymmetric monomer side groups do exist, only those polymers that exhibit syndiotactic or isotactic character are considered. There is still another type of polymer system, the block copolymer, which can exhibit limited crystallization. This is a polymer whose monomer units A and B are jointed together in such a fashion as to pro- * .. If the blocks of duce a polymer of the type A-A-A-B-B-B-AAs are the same size and if the same is true of the blocks of Bs, and if the repetition of A and B blocks occurs in a regular fashion, then crystallization can occur in the system. If there are long runs of A or B blocks, one of which is crystallizable, crystallization can also occur. In actual fact, for a large number of reasons, no polymer system is completely crystallizable. At best, a conglomerate of crystalline and amor-

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GENERAL BACKGROUND ON SEMICRYSTALLINE POLYMERS

34 1

phous regions are produced. It is this semicrystalline polymer system that we will discuss in this part. For a more extensive description of polymer crystallization requirements, the reader is referred to Khoury and Passagha,' Geil,2 and W ~ n d e r l i c h . ~ 10.2.2. Historical Development 10.2.2.1. Early Experiments and Resulting Models. In the case of polymers crystallized from the bulk (i.e., from their own melts), the predominant morphology that is observed using a polarizing microscope is spherulitic (Fig. 1) in nature. This type of structure has also been observed in simple compounds and minerals. The extinction patterns produced, i.e., the familiar maltese cross or the alternation of light and dark concentric rings, remained unchanged as the sample was rotated between

FIG. 1. Transition from spherulitic to axialitic morphology in linear polyethylene ( M , = 30,600, M , / M , = 1.19) with decreasing undercooling (optical micrographs, crossed nicol prisms).

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FIG. 2. (A) Polyethylene spherulite with a schematic of the radiating chain-folded lamellae. Twist of lamellae causes the characteristic dark rings and patches, depending on the regularity of twist. (B) Schematic of the fringed micelle showing crystalline and amorphous regions. (C) Model for surface nucleation and growth of a regularly chain-folded crystal with adjacent reentry. The crystal grows in the G direction by the addition of chains having thickness b and width a . The thickness of the crystal itself is denoted as 1 (the initial thickness prior to annealing is called / g * ) . The lateral and fold-surface free energies are u and re, respectively. (D) Schematic showing possible departures from strictly regular folding as well as possible causes for defects in the body of the crystal (Fig. 2B-D from Hoffman et a/.").

the polarizing and analyzing elements of the microscope, indicating the symmetry of the spherulitic structures. The nature of the long but narrow fibrillar crystals making up the spherulite (Fig. 2A) was not known for quite some time. Indeed the precise nature of such structures still remains a source of controversy and a reason for continued research. Bunn and c o - w ~ r k e rwere s ~ ~responsible ~~~ for several important observations concerning polymer crystals grown from the melt, particularly the crystals found in linear polyethylene. It was known from X-ray line

la

C. W. Bunn, Trans. Faraday SOC. 35, 482 (1939). C. W. Bunn and T. C. Alcock, Trans. Faraday SOC. 41, 317 (1945).

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broadening measurements that the polymer crystals were quite small, and also from Bunn it was learned that the polymer chains of the crystallized molecules were perpendicular to the radial direction in the spherulites. In the same studies Bunn demonstrated that the unit cell of crystallized polyethylene did not change as a function of molecular-weight distribution. Finally, he showed that the polymer crystals exhibited a high degree of perfection and, in particular, that the unit cell of crystallized bulk polyethylene was similar to that of the orthorhombic form of the normal alkanes. These observations, coupled with the knowledge that the crystals were of the order of a few hundred Angstroms thick, led to the postulation of the fringed micelle model (Fig. 2B). In this model the polymeric system was viewed as an amorphous, random-coil matrix, with bundles of extended chains of adjacent molecules forming crystalline domains. In such a model any given molecule could contribute to one or more of the crystalline domains, remain completely in the amorphous region, or in fact contribute to both the crystalline and amorphous regions. The amorphous matrix was visualized as being noncrystalline by virtue of atacticity , shortness of molecular length, entanglements, etc. The fact that a single molecule could be a part of more than one crystalline domain was of great usefulness in explaining some of the mechanical properties of polymers, and for quite some time the concept of the fringed micelle was the accepted one. Later, however, certain new experimental evidence began to cast doubt upon the validity of this model. It has been shown that single crystals of certain polymers could be prepared by crystallizing them from solution. l6 One of the worrisome details associated with the fringed micelle model was that as the crystal grew in size there naturally accumulated a large amount of strain at the ends of the crystal where the molecule emerged again into the amorphous matrix. Since this strain would be cumulative, the crystals by necessity would be limited in “girth.” But now there was evidence that, at least from solution, single crystals of appreciable size could be prepared. Finally, in 1957 Keller“ showed that polyethylene could be crystallized from a dilute xylene solution in the form of singlecrystal platelets of the order of 130 A thick, with electron diffraction patterns indicating conclusively that the polymer chains were perpendicular to the flat surfaces of the single crystal. It became apparent,17-19 since the length of the average polyethylene molecule was many times that of R. Jaccodine, Nature (London) 176, 305 (1955). A. Keller, Philos. M a g . [S] 2, 1171 (1957). la E. W. Fischer, Z . Naturforsch., Teil A 12, 753 (1957). lo P. H. Till, Jr., J . Polym. Sci. 24, 301 (1957). l6 l’

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the 130 8, found for the crystals, that a single molecule must fold back into the crystal many times, the number of folds being limited only to the molecule’s length. At the same time other investigations were showing through the use of electron microscopy and X-ray techniques that, even in bulk crystallized polymers, the same type of structures were present, i.e., radiating lamellar crystals wherein the molecules were perpendicular to the flat planes of the lamellae, which were found to be of the order of 50-150 A thick. It is interesting to note that as early as 1938, one investigator seriously considered a folded crystal as being a good model from the available experimental evidence.20 In the 1950s, when this new information regarding polymer solution-grown single crystals became available, several models involving chain folding were proposed. The most simplistic model was that of a crystal composed of polymer molecules with exclusive adjacent reentry of folds (Fig. 2C), resulting in a regularly folded surface that, with the tilted structure of the four-sectored single crystals, resembled a terra cotta roof. The only defects were the ends of the molecules, which either formed as short-length flexible rods or cilia above the folded surface or else folded back into the crystal. Of course, the crystal had to accommodate certain defects such as small degrees of atacticity in the molecule body or ends, the problem of matching ends of molecules, and multiple nucleation of the same molecule. At the other extreme is the model that was proposed by Florya21 In this model it was postulated that folding exists, but that the molecule reenters the crystal somewhat randomly. The fold surface would be very rough with loops of various lengths due to this random reentry, entanglements, etc. This model was termed the “switchboard” model in analogy to the old telephone switchboards and their entangled or over-lapping plug-in leads. Despite the details of the,controversy, the question still arose as to whether any model that explained the relatively perfect sectorized solution-grown single crystal could be applied to crystallization in the bulk. Over the intervening years from the time of Keller’s first monumental work, many capable investigators have used a variety of techniques in an attempt to conclusively demonstrate the true nature of the polymer crystals in bulk. In particular, infrared spectroscopy, small-angle light scattering, nuclear magnetic resonance, and neutron scattering have been employed. The results of these studies are again not conclusive in providing a unique model consistent with all the experimental observations. *O

21

K. H.Storks, J . A m . Chem. SOC. 60, 1753 (1938). P. J. Flory, J . A m . Chem. SOC. 58, 2857 (1962).

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The general structure of spherulites and axialites (open-structure spherulites) is now thought to be due to long, ribbonlike lamellar, chain-folded crystals, which exhibit small-angle branching as they grow (Fig. 1). Electron-microscopic investigations have clearly shown that the fibrils or lamellar crystals rotate or twist about their long-growth axes (Fig. 2A). A number of investigators notable among whom were Keller,22 Point,23Keith and Padden,24*25 and Pricez8have observed and shown how this crystal rotation about the radii of the growing spherulite is responsible for the characteristic extinction patterns observed in the spherulites. In the classical work of Keith and Padden,27-2sthey were able to provide a phenomenological explanation of spherulitic growth in polymers, the basic problem being to explain how the growth front degenerates into a group of fibrils. In polymers it is recognized that the growth kinetics result not from diffusion of material away from the growing interface, but rather from a nucleation process occurring at the tip of the lamellar crystal. Keith and Padden conclude that the morphological character of the growing spherulite is controlled by the rejected material concentrated at the growth front. In their terms the impurity produces cellulation wherein there is produced, at the growth front, fibrils having a width S = D / G , where 6 is of the order of 100 nm, D is a diffusion constant, and G is the growth rate. The theory proposed by Keith and Padden explains the branching and ribbonlike character of the lamellar crystals and, in addition, provides a mechanism for the fractionation of material as a result of the crystallization process. Materials unused in the initial crystallization process are deposited between growing fibrils. This rejected portion may crystallize much more slowly, may crystallize at a lower temperature, or may not crystallize at all. The explanation still allows the radial growth rate to be a constant as more and more of the liquid nutriment is used until such time as neighboring spherulitic structures impinge. Keith, Padden, and Vadimsky30have performed a series of crucial experiments showing the existence of tie molecules and/or crystallized columns of molecules (probably crystallized in an “extended-chain” structure) holding together the ribbonlike lamellae. They performed a series of A. Keller, J . Polym. Sci. 39, 151 (1959). J. J. Point, Bull. Cl. Sci., Acud. R . Belg. [ 5 ] 41, 974 (1955). 24 H. D. Keith and F. J . Padden, Jr., J . Polym. Sci. 39, 101 (1959). 25 H. D. Keith and F. J . Padden, Jr., J . Polym. Sci. 39, 123 (1959). 26 F. P. Price, J . Polym. Sci. 39, 139 (1959). 27 H. D. Keith and F. J . Padden, Jr., J . Appl. Phys. 34, 2409 (1963). H. D. Keith and F. J. Padden, Jr., J . Appl. Phys. 35, 1270 (1964). 29 H. D. Keith and F. D. Padden, Jr., J . A p p l . Phys. 35, 1286 (1964). 30 H. D. Keith, F. J. Padden, Jr., and R . G . Vadimsky, J . Appl. Phys. 42, 4585 (1971). 22

23

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A N D CRYSTALLIZATION

experiments wherein polyethylene was diluted with dotriacontane. The n-alkane was subsequently leached out of the crystalline polyethylene and a series of electron micrographs clearly showed the existence of large numbers of these intercrystalline links between the lamellae. The theory does not explain the reason for the twist in lamellae as they grow. However, the general details of lamellar crystals connected through single molecules entering into the crystallization of two or more chain-folded lamellae, is an accepted fact. The bulk of the experimental and theoretical evidence supports the idea of adjacent reentry with relatively uniform folds. The questions of how much adjacent reentry occurs and the tightness of the fold loops, etc., is still not answered conclusively. 10.2.2.2. Theoretical Basis for Crystallization Studies. The major thrust of this part will be devoted to an understanding of nucleation and crystallization as it applies to the bulk polymer systems. The view we choose to take here is that in such systems the primary nucleation process is heterogeneous, i.e., the formation of a stable nucleus as aided by foreign particles or surfaces. In general, such a nucleus is envisioned as being of a folded nature, similar to that of the crystal lamella that will be produced. This ribbonlike lamella will have a fold distance, i.e., the distance from the top folded surface to the bottom folded surface, which is dependent upon the temperature of crystallization (Fig. 2D). The fold surface will be rough, implying that some of the folds will be “floppy,” that there will be extending ends or “cilia” protruding above the fold surface, that there will be some molecules or portions of molecules adsorbed on the fold surface, and that there be tie molecules between one fold surface of one lamella and the fold surfaces of adjacent ones. However, the fold surfaces will show a marked preference for adjacent reentry and a general regularity of fold. The fact that crystals have an amorphous content has required the concepts of adsorbed molecules, cilia, tie molecules, etc., between adjacent lamellae to be introduced. In addition, even in the most crystalline of the semicrystalline polymers, there will be a substantial percentage of the total molecular population that does not crystallize in this initial crystallization process or in fact does not crystallize at all. Recent developments in etching and staining of samples for the electron microscope reveal a wide variety of crystallized entities in the bulk samples.31 In addition to the major lamellae discussed above, there also exists crystallized material whose top to bottom measurements are much smaller than the fold distances associated with the larger lamellae. Such structures primarily reside between the larger lamellae and may or may not be folded. They certainly represent material that has crystallized at a later time. Finally, because of lack of required stereo regularity, or entan31

D. C. Bassett and A. M. Hodge, Pruc. R . Soc. London. Ser. A 359, 121 (1978).

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347

glement, molecular size, etc., a significant portion of the polymer does not crystallize. The primary crystallization process, nucleation controlled, consists of the formation of these lamellae, radiating outward and branching in such a fashion that the overall crystal density of the growing structure remains nearly constant. In some cases the ribbonlike lamellae become twisted as they grow, giving rise to the spectacular ringed spherulites where variations of this rotating nature accounts for the production of ruffles and rings in the spherulitic pattern. Again, both the outward radial growth rate and the lamella thickness is a function of the growth temperature. The magnitude of the radial growth rate is itself strongly dependent upon the molecular weight of the growing species. The descriptions of the techniques used in the measurement of such radial growth by either observing a few spherulites separately or by observing the bulk growth changes resulting from many such spherulites will occupy the major portion of this part. In respect to the material covered herein, we first discuss bulk crystallization and then crystallization from solution. As far as the development of crystallization theory, and specifically as it applies to chain folding, the study of the growth of single-polymer crystals from solution has immense historical importance. The exact nature of such crystals is still involved in a bit of controversy, but for our part, we envision crystallization from solution as, under carefully controlled conditions, producing chainfolded, single crystals having a high degree of regularity. The folded surfaces will have some degree of roughness caused, in general, by the same mechanisms that caused the roughness and amorphous nature of the surfaces between the lamellae produced in bulk crystallization. Considerably less detail will be given to two other important crystallization processes, namely, crystallization induced by hydrodynamic flow (Pennings’ shish kebob type of crystals), and crystals produced by the application of high pressures. The latter produces crystals having extremely large fold distances, which may be viewed as being much closer to extended-chain crystals than to the folded ones. The former may be viewed as being composed of a central filament composed of nearly extended-chain crystallized material upon which at somewhat regular intervals there appear buttons composed of folded crystals with the fold surfaces perpendicular to the central filament. As is the case with all other viable experimental fields there is a great deal of controversy associated with the exact explanations of such growth. 10.2.2.2.1. THE AVRAMITHEORY.Avrami published two paper^^^,^^ 32 33

M . Avrarni, J . Chem. Phys. 7 , 1103 (1939). M. Avrarni, J . Chern. Phys. 8, 212 (1940).

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10. NUCLEATION A N D CRYSTALLIZATION

relating to the kinetics of phase change in general. His fundamental relationship as applied to the crystallization process was

x = 1 - exp(-Ktn),

(10.2.1)

where x is the degree of crystallinity, K the crystallization rate constant, t the elaspsed time, and n an integer characteristic of the type of nucleation. In Avrami’s development and the extension of this work by Morgan,34n may take on values of 1, 2, 3, and 4,depending on the nature of the nucleation and growth process. For spherical growth n is 3 or 4; for platelike growth n is 2 or 3; and for fibrillar growth n is I or 2. Later it will be shown that for some polymers spherulitic growth may exhibit an n of 3 of 4, depending upon whether or not nuclei appear sporadically (homogeneous nucleation) in time or are in existence from the beginning of the crystallization process (heterogeneous nucleation). With other polymeric systems it will be shown that n can exhibit values from less than l to more than 4, these experimentally determined values not necessarily being integers. The complexity of the problem arises from the fact that there may be two or more processes taking place simultaneously, that n may be a function of the degree of crystallization, and that the degree of crystallization itself is not well defined. However, the use of the Avrami relationship can produce a much needed insight to the understanding of the crystallization process. The rate constant K in the Avrami equation is equal to 4rvoG3/3, and n = 3 when the spherulites formed were “born” at the same time. We can use the relationship to determine G, the radial growth rate, providing we know the number of heterogeneous nucleation centers per unit volume, vo. 10.2.2.2.2. KINETICGROWTHTHEORY.Providing the spherulites are formed through the process of heterogeneous nucleation, the primary nucleus is developed from a solid surface such as a dirt particle. It is then postulated that additional molecules are laid down and become a part of the chain-folded lamellar structure by first having a portion of the molecule form a stable folded nucleus on the existing crystal substrate. Following the formation of such a nucleus the rest of the long-chain molecule is “reeled in,” crystallizing with a fixed, temperature-controlled fold distance. This type of nucleation-controlled growth seems very consistent with our knowledge of the fibrillar growth of the spherulites. Applying the theoretical idea of Turnbull and Fisher,35the concept of formation of L. B. Morgan, Philos. Trans. R . Soc. London, Ser. A 247, 13 (1954).

s5

D.Turnbull and J. C . Fisher, J .

Chem. Phys. 17, 71 (1949).

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349

critical-sized nuclei can be expanded to give the following growth relations hip:

G = Goexp

[

-

g] [

exp - K ,

1 (10.2.2) (m)],

where AF* is the activation energy for transport of segments to the site of crystallization, T the temperature of crystallization, k the Boltzmann constant, and AT = Tmo - T , where T , is the melting point of a defect-free, large extended-chain crystal of the polymer. With the use of a reasonable approximation for A F * , it is possible to compute the growth rate constant K , . Using the flux equation approach Hoffman36 calculated this rate constant to be 4houueTmo/kAhf in one limit (see later). In this relationship Ahf is the heat of fusion, 6 0 the monomolecular layer thickness, and (+ and u, the lateral surface and fold surface free energies respectively. Optical microscopy can be used to determine radial growth rates, and Eq. (10.2.2) allows observations made from measurements on single spherulites to be compared with the calculated growth rates from such experimental techniques as dilatometry , which provide a measure of the bulk crystallization rates, Eq. (10.2.1.). Relying on these two relationships, we can calculate the free-energy product from the computed rate constant. 10.2.2.2.3. HOMOGENEOUS NUCLEATION THEORY.The measurement of the growth rates allow one to determine values of (+re. If one then determines the value of v2(+,from homogeneous nucleation experiments, the values of both (+ and U, can be determined. The generalized equation for the homogeneous nucleation rate I is defined as

I (nuclei ~ e c - k m - ~ ) = I. exp[- ( A F * / k T ) ] exp[-K,(l/T A T 2 ) ] , (10.2.3) , is where, according to Fisher and T ~ r n b u l lI,, ~=~ ( N / v m ) ( k T / h ) which approximately equal to for materials such as polyethylene; N is Avogadro’s number; 7, the molar volume of a sequence of chain segments whose length is approximately that of the crystal nucleus; k T / h the usual frequency factor, where k is Boltzmann’s constant, T the temperature, and h Planck’s constant; AF* the activation energy for transport of crystallizing molecules; and K i is the homogeneous nucleation rate constant. K i is defined by Lauritzen and Hoffman3’ as 30.2u2(+, ( T m 0 ) 2 / K ( A h f ) 2ATP f , where Tmo, k , A h f , and AT have the same 38 37

J. D. Hoffman, SPE Trans. 4, No. 4, 315 (1964). J. I . Launtzen, Jr. and J. D. Hoffman, J . Res. Nor/. Bur. Stand.. Sect. A 64,73 (1960).

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meanings as in Eq. (10.2.2.). The factor 30.2 is obtained when it is assumed that the homogeneous nucleus builds upon the orthorhombic unit cell. Because of the very large undercoolings encountered in homogeneous nucleation experiments, the thermodynamic driving force must be corrected by a factorf, which is equal to 2T/(Tm - T ) as demonstrated by Hoffman and Hoffman et al. 39 have shown that it is reasonable to assume that cr and cre vary with temperature and suggested a dependence of the form

( 10.2.5)

10.2.2.2.4. THEORYRELATING TO Tmo. As an adjunct in determining the surface free energies from the growth and nucleation constants, Ahf and Tmoare necessary input information. In the case of simpler materials, the experimental determination of the heat of fusion and the melting points from the known crystal form is straightforward, using timehonored physical methods. However, in the case of semicrystalline polymers the determination of either quantity is not easy. From the description of “semicrystalline,” it is obvious that accurate heats of fusion cannot be obtained by a simple determination of the melting heat from any given sample. This problem is discussed in Part 9 of this volume. It is sufficient here to state that the problem is not a trivial one. The determination of the quantity Tmois also discussed in Part 9. However, we briefly discuss the problem here. We have previously defined Tmoas being the melting temperature of a large perfect crystal in an extended-chain configuration. Obviously, it must also be the same crystalline form as the bulk crystal. Extended-chain crystals are not easily obtained experimentally. The usual form of crystals in bulk polymers is that of chain-folded lamellae, but the experimental melting point from such crystals is severely depressed by the smallness of the fold distance (i.e., the thinness of the crystal). W ~ n d e r l i c hhas ~ ~shown that for linear polyethylene, crystallization under conditions of high pressure produces crystals that in the extreme may closely resemble extended-chain crystals. Experimental melting points from such materials are indeed much higher than those obtainable from crystals of the same material grown under atmospheric pressures. In certain cases an extrapolated J. D. Hoffman and J. J. Weeks, J . Chem. Phys. 37, 1723 (1962). J. D. Hoffman, J. I. Lauritzen, Jr., E. Passaglia, G . S. Ross, L. J. Frolen, and J. J. Weeks, Kolloid-2. & Z . Polym. 231, 564 (1969). B. Wunderlich and T. Arakawa, J . Polym. Sci., Part A 2, 3697 (1964). 38

38

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GENERAL BACKGROUND ON SEMICRYSTALLINE POLYMERS

35 1

melting point has been computed from data obtained from experimental melting points of simple molecules, which may be considered as precursors to the high-molecular-weight polymers, e.g., the use of such properties of the linear normal alkanes to predict the melting points of polyeth~ ~has been shown that the ylene by Flory and Vrij4I and B r o a d h u r ~ t . It fold-to-fold distance of crystallizable polymers is a function of the temperature of crystallization. It has also been shown that when such crystals remain at the crystallization temperature, this fold-to-fold distance 1 increases as a function of time, the increase having the general form of l ( t ) = 12

+ B log f.

(10.2.6)

It has further been shown that when crystals grown at a lower temperature are raised to and remain at a high temperature, again there is an increase in the fold-to-fold distance. By Eq. (10.2.6.) it is seen that the thickness (fold-to-fold distance) at any time, i(t),increases from the initial thickness 12 logarithmically with time. In general, this relation can be simplified 1 = yl2.

(10.2.7)

This equation then presumes that at some final time, the initial thickness times the constant y will give the final fold distance. With certain other simplifying assumptions it can be shown that ( 10.2.8)

where T , is the experimentally observed melting point obtained by crystallizing at T , and allowing the crystals to remain at T , for sufficient time as to change 12 by a factor of y. Then a plot of T , vs. T , can be extended linearly until intersection with the line T , = T , to obtain Tm0,the melting point of the large extended crystal. There will be different lines corresponding to different y values, but all will intersect at the same temperature. It is therefore presumed that no further thickening occurs during the melting process. t P. J. Flory and A. Vnj, J . Am. Chem. SOC. 85, 3548 (1963). M. G . Broadhurst, J . Res. Natl. Bur. Stand., S e c t . A 70, 481 (1966). J. D . Hoffman and J. J. Weeks, J . Res. Nut/. Bur. Stand., S e c t . A 66, 13 (1962). 1p J. D. Hoffman, G . T. Davis, and J. I. Lauritzen, Jr., Treatise Solid Stnte Chem. 3, 497 (1976). 41

42

t Experimental measurements of T,,, vs. T , for a given polymer are much easier and less time consuming than direct measurements of I by electron microscopy or small-angle X-Ray diffraction. However, the experimenter should be aware that the value of y may be dependent on both time and temperature, resulting in errors in the determination of Tm0for some systems. For a further discussion of problems in the use of either Eq. (10.2.8) or (10.2.9)the reader is referred to Hoffman et a/.,44p. 528.

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An interesting melting point relationship that involves fewer assumptions and that has a straightforward theoretical basis is the following:43

T , = Tm0[l - 2ce/(Aht)l].

(10.2.9)

Since by plotting T , vs. 1/1 one obtains an intercept of Tm0and a slope of - 2ae Tmo/Ahf,this single relationship not only allows calculation of the required melting point, but in addition provides a value for ve if Ahf is known. Again, one of the chief experimental problems is assuring that 1 does not change during measurement of length or melting point. One group of investigator^^^ used moderate gamma-ray dosage to induce cross-linking. They measured both dissolution temperature and the melting point for polyethylene single crystals. The effect of the radiation has received some comment, but the technique as used by these and other observers seems to be a reasonable compromise solution to the problem of thickening while making the required measurements. The usual methods used to measure 1 are small-angle X-ray measurements and direct length measurements using electron microscopy. The whole problem of determining the melting point of the infinite polymer crystal has been developed in Part 9 of this volume. In this part we restrict our remarks to some experimental work that has been hone in our laboratory using differential thermal analysis.

10.3. Experimental Methods for Measuring Crystallization Ratest 10.3.1. General Considerations

There are several hundred papers in the literature that report the study of crystallization in a wide variety of polymers using a large number of different methods. In general, there is a lack of experimental detail with regard to the characterization of the actual samples and the actual techniques used (both with regard to equipment and handling techniques); in many cases the temperature control of the system is poor. It is hoped that this section will provide the reader with some insight regarding the care needed in performing experiments using a variety of methods. The experimental observations of the development of crystallinity in polymer samples fall into two groups. The first type of measurement is that of direct observation of the development of spherulites or single crystals under isothermal conditions. The methods used in this type of 45

H. E. Bair, W. Huseby, and R. Salovey, J . Appl. Phys. 39, 4969 (1968).

t See also different chapters in Vols. 6A and B of this series,

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353

experiment are optical microscopy, transmission electron microscopy, and some small-angle scattering techniques. These experiments require a sample that is prepared in the thin-film configuration. The second type of observation consists of following the development of the total crystallinity of a sample under isothermal conditions. This type of experiment requires the measurement of a physical property which is very sensitive to changes in crystallinity such as specific volume or density, infrared absorption bands, X-ray intensities, dielectric properties, and thermal properties. These methods include dilatometry, density balance and density gradient methods, differential thermal analysis, and various infrared, X-ray scattering, and dielectric techniques. These techniques usually require a sample that is in what we shall call the bulk configuration. Before we become involved with the details of sample preparation, there are several general considerations to be discussed. In addition to having a distribution of molecular weights that influences the primary nucleation and subsequent crystallization, a typical polymer has many other nonpolymeric materials that have been deliberately added or inadvertently found their way into the polymer. Examples of deliberately added materials include plasticizers, antioxidants and other stabilizers, other polymeric materials, and solid fillers of various types. Material left from the polymerization or subsequent working of the sample as well as dust or other solid debris are examples of the types of impurities that are added to the sample inadvertently. Before one decides how best to prepare the sample for study, the scientific purpose of the experiments must be determined. If, in fact, the purpose is to study the “as is” whole polymer then, although thorough characterization may be required, the study should be performed on the material as received. If, however, information of a more fundamental nature is desired, then a detailed study of the effect of molecular weight and molecular-weight distribution on crystallization is needed without the sometimes enormous but usually unknown effect of the foreign materials. In the latter case, fractionation, characterization, and -a general “cleaning up” procedures are necessary prerequisites to the actual crystallization study. In terms of time, cost, and scientific ingenuity, this step is often the most demanding one. Although some polymers can be synthesized in such a way that the resulting polymer is nearly monodisperse (e.g., the anionic polymerization of polystyrene), usually the polymerization procedure results in an uncontrolled mixture of molecular weights ranging from the terminated monomer to molecular weights ranging into the millions. The most widely used methods of fractionation at the present time are column elu-

354

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NUCLEATION A N D CRYSTALLIZATION

tion techniques and gel permeation chromatography (GPC). The determination of the molecular-weight distribution is most easily done by GPC if such a calibrated device is available. In addition to GPC the classical methods of viscometry, osmometry, light scattering, or ultracentrifugation may be used to determine molecular weights as well as molecularweight distribution (see Part 2 in this volume, Part A, for detailed discussion of these techniques). 10.3.2. Methods Using a Thin-Film-Type Specimen? 10.3.2.1. Sample Preparation. A thin film of polymer (of the order of 20-50 pm thick) sandwiched between two thin transparent covers is perhaps the most popular sample configuration when direct observation of crystal growth is required. This technique was used extensively by the authors in a study on the growth kinetics of polyethylene. Although the details described here refer directly to that work, the descriptions are quite general in nature and apply directly to other polymeric materials. In our study on polyethylene, we wished to prepare samples that contained a small number of nucleation centers so that individual spherulites could be observed for longer periods of time prior to impingement with neighboring spherulites. Further, we wished to use the same field for all of the observations on a single sample. In some materials, the sample degrades at the temperature of the experiment and this effect can be detected by irreproducibility of the results on subsequent runs. When this is the case a new sample must be prepared prior to each run. In our work, the samples were cleaned according to the flow chart shown in Fig. 3. The filtering techniques used were similar to those used for preparing a sample for light-scattering studies in solution. Regenerated cellulose filters of controlled pore size were used throughout the experiments. The filtration was done in a stagnant air box and the solution was filtered very slowly (about one drop per second). Faster filtration rates resulted in less clean solutions. Once the filtering was complete the sample was collected on a clean filter and dried at 80°C under vacuum to assure that all solvent had been removed. Part of this sample was used to characterize the material and the rest of the sample was used to prepare the thin-film specimen. Between 100 and 300 mg of the sample was prepared for viewing by lightly pressing the molten sample between two 1-in. square microscope cover glasses. This was done at 150°C in a vacuum oven. The use of spacers between the cover slips allowed the preparation of samples of unit See also Volume 6A, Chapter 2.5, as well as Volume

1 1 in this series, Part 12.

F I L T E R XYLENE 0.2 p F I L T E R

DISSOLVE PE FRACTION I N XYLENE ( . l % by w e i g h t )

MEASURE MOLECULAR WEIGHT D I S T R I B U T I O N BY GPC

T

VACUUM

FIG.3. "Preparation" of polyethylene samples for growth studies.

356

10. NUCLEATION A N D CRYSTALLIZATION

form thickness. The thin film should nearly cover the entire surface of the cover glasses. The cleaning pretreatment of the cover glasses is very important since surface nucleation of the sample is not desirable. In our studies the cleaning method used was as follows: (1) cleaning with hot chromic acid solution, (2) rinsing with distilled water, and finally (3) further cleaning in distilled water using ultrasonic scrubbing techniques. After complete drying in a vacuum, cover slips treated in this way were found to cause little or no surface nucleation in the polyethylene samples. In the case of polyethylene, it was found that although some thermal or oxidative degradation occurred at the edges of the polymer film, the usual temperatures employed during the melting and crystallization sequences caused no noticeable changes in the crystallization kinetics of the central portion of the polymer film. Measurement of the molecular-weight distribution in the central portion of the film, using GPC showed no change in the distribution even after heating the sample to 150-155°C for as long as 200 hours. The choice of the thickness of the sample to be employed is dictated by the experimental method used. In the case of optical microscopy, the optical amplification most generally used is of the order of 3 0 0 ~ . In this case the smallest crystal aggregate that shows any detail is approximately 5 pm in its largest dimension. The field of observation is approximately 500 p n in diameter. Even when there are only a few active nuclei in the field, by the time the spherulites have grown to approximately 20 pm one or more of their surfaces must impinge on the glass cover slips. Thus, for most of the growth process one observes growth that is restricted to two dimensions. Although one would like to eliminate this problem by using thicker samples, the numerous nucleation centers that are scattered through the specimen in proportion to the sample thickness cause rapid overgrowth, which makes accurate measurements impossible. 10.3.2.2. Optical Microscopy. Two methods are available for measuring the growth rates of polymer crystals using the optical microscope. The first is the growth in situ on the stage of the microscope using a suitable hot or cold stage. This method has one major disadvantage; namely, a microscope and the auxiliary equipment can be tied up for weeks or even months on a single experiment. However, it offers the tremendous advantage that individual spherulites can be observed from their first development up to the time when impingement occurs. The second method consists of preparing a large number of samples, which can be crystallized away from the microscope, for instance, in a temperaturecontrolled bath. In this technique the samples are heated to a temperature above the melting point for a period of time, and then placed in another bath, which is set at the desired crystallization temperature. The

10.3.

MEASURING CRYSTALLIZATION

RATES

357

samples are removed from the bath at specified time intervals and quenched rapidly to a lower temperature. Self-decoration occurs and the size of each spherulite can be measured with a microscope as a function of residence time at the given crystallization temperature. This method has the advantage that equipment is available for other uses while the experiment is in progress, but has the disadvantage that sample-to-sample variations can occur and each spherulite can be examined only once. For a complete study on a given material a combination of the two methods is recommended. 10.3.2.2.1. TEMPERATURE-CONTROLLED STAGES.Regardless of the particular technique employed, the most obvious prerequisite is that there be a temperature environment with the capability of constant temperature control for long periods of time. For a polymer like linear polyethylene, the experimentally accessible temperature range may only be from 3 to 6", but over this range the growth rate will change by a factor of 1000. To make accurate growth measurements, it is necessary that the temperature bath or hot-cold stage should vary by no more than +O.O5"Cfor periods of time up to several weeks in duration. There are several constant temperature baths available commercially that will easily satisfy this specification. However, there are to our knowledge no microscope hot-cold stages commercially available that satisfy this requirement. In our own work we made stages that controlled the sample environment to within & O.O2"C, and where the sample temperature was accurately measured to +O.Ol"C. There are many proportionating temperature controllers commercially available that have the ability to sense and control temperature to 2 0.005"C, using either thermocouple junctions or thermistor sensors. The major problem in constructing the hot-stage environment is that of spatial dimensions, particularly the limitation of only a 2 cm clearance between the stage platform and the objective. Four our studies of the crystallization of linear polyethylene, we chose to use hot stages that were electrically heated, and two types were constructed. For those experiments in which only constancy of temperature was a prerequisite, we constructed stages having a high thermal mass. For those experiments demanding quick changes in temperature as well as eventual constancy, we constructed stages having such small thermal inertia that the thin fdm of sample itself was the limiting thermal mass. In both cases a square box having external dimensions 12.5 x 12.5 x 2.5 cm was constructed to house the sample and heating environment. The box was constructed of balsa wood planks 0.80 mm thick with sheets of aluminum foil laminated between them. Three balsa wood layers and two aluminum foil layers were used to construct the top, bottom, and sides of the box. The

358

10. NUCLEATION A N D CRYSTALLIZATION

layers of balsa were cross-grained from layer to layer and the whole sandwich was held together with silicone cement. Even though the temperature of the sample inside was above 150"C, the outer temperature of the top or bottom of the box was less than 40°C. A slow flow of room temperature air was passed between the condenser and the bottom of the box as well as between the top of the box and the objective. This air flow was sufficient to prevent any appreciable rise in temperature of the optics. While externally similar, the two types of cells were very different. For the low-thermal-inertia cell, the thin film of sample was positioned between two parallel grids each 10 x 10 cm. These heating grids were composed of a large number of taut parallel 0.025 mm diameter Advance metal heating wires, each wire being separated from the next by approximately 0.05 mm. Each wire was attached at its ends to a brass strip and the entire grid network held taut and parallel by springs. The sample itself was, of course, in the focal plane of the microscope being held and positioned by a micromanipulator attached to the metal stage of the microscope. The two heating grids above and below the sample were sufficiently far out of the focal plane as to remain invisible when the sample was observed through the microscope optics. A small bead thermistor on the sample was used as the temperature controller sensor. By far the largest thermal mass was in the sample itself, and the system was capable of exceedingly rapid temperature changes while still capable of excellent constancy of temperature when desired. In the other type of cell, relatively thick copper plates were used as a support for electrically insulated heating wire. The wire was in grooves cut into the copper plates, and the temperature sensing and control thermistors were embedded in the top surface of the bottom plate, next to the sample. Rapid temperature changes were impossible with this cell, but the constancy of temperature was very good. In both cases copperconstantan thermocouple junctions were used to measure the sample temperature. The temperature of the sample was continuously recorded using thermocouples that had been previously calibrated. A suitable reference such as a water-ice bath is needed. In our case, a water triplepoint cell was used. In both types of cells the two heating grids were electrically connected as a parallel resistive network, with a variable trim resistor between the two. The control sensor was always near the bottom grid, and the trim resistor was adjusted so as to have both grids at precisely the same temperature when operating at constant temperature. A differential thermocouple pair ofjunctions was used to make certain that the top grid had the same temperature as did the controlled bottom one. When using the low-thermal-mass cell, the sample temperature could

10.3.

MEASURING CRYSTALLIZATION RATES

359

be lowered in a controlled fashion at rates in excess of 10”per minute. Of course, the temperature could be raised at a much faster rate. With the cell having high thermal mass, the sample temperature could only be dropped at a rate somewhat less than 1” per minute. The sample temperature could be raised more rapidly but still at a much smaller rate than that obtained with the previously described cell. 10.3.2.2.2. THE“BATH” TECHNIQUE. There are several advantages to using the constant temperature bath over using an in situ microscope hot-cold stage: 1. The method does not require continuous use of a polarizing microscope or camera equipment. The measurements of the spherulite or axialite dimensions can be made at one’s own convenience. Even if a camera is employed it too is only in use for short periods of time. 2. The method requires many samples. Therefore, possible sample to sample variations are considered in the resulting growth statistics. 3. Because of the large thermal mass of the bath, short-term electronic variations or even complete power outages of a very short duration can be tolerated. With the hot stages any controller difficulties are almost immediately reflected by a change in temperature of the sample. 4. The shorter periods of time required for thermal equilibrium make it possible to determine the crystallization rates at larger undercoolings. 5 . It is easier to employ the self-seeding technique. However, there are also certain disadvantages in using the constant temperature baths:

1. Much larger quantities of sample are required. 2. Three baths are needed: one to melt the sample, one to crystallize the sample, and one to quench the sample. 3. It must be assumed that all spherulites commence to grow at the same time. 4. There is greater uncertainty with regard to the actual periphery of the growing crystal, i.e., measurement errors are usually greater. This technique is relatively simple and straightforward. The thin-film sample specimens are melted and then placed directly in the constanttemperature bath. A continuously running timer is used so that the removal of each sample can be chronologically logged. At appropriate time intervals a sample is removed from the bath and quickly quenched in to order to decorate the growth periphery by rapid crystalization producing a different morphology or by quenching the remaining liquid to a glass. The time that the sample was in the crystallizing bath is noted, and at the experimenter’s leisure, the dimensions of the spherulites can be deter-

3 60

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NUCLEATION A N D CRYSTALLIZATION

mined. Wherever possible, radial measurements should be made, several for each spherulite or axialite. By this technique, the entire area of the sample can be surveyed and a large number of measurements made. A plot of the average radial growth vs. time of crystallization should be linear. The intercept on the time axis is the induction time and will be zero if crystallization commenced immediately after temperature equilibrium occurred. The linearity of the growth vs. time plots is usually good providing that no measurements are made of spherulites that are close to impinging upon each other. 10.3.2.2.3. THE in Situ TECHNIQUE. This technique is similar to that employed with the constant-temperature baths, except that only one sample and one thermally controlled hot stage are used. The sample is melted by raising the temperature of the cell. After melting, the cell temperature is lowered to the desired crystallization temperature. In a relatively short period of time the cell is in thermal equilibrium and crystallization commences. If the type of cell having a large thermal mass is used, it is better to preset the cell temperature to the desired crystallizing temperature and to melt the sample using another thermal environment such as a hot bar. The sample is then inserted into the controlled cell and attached to the micromanipulator. With cells of low thermal mass the change in temperature from that required for melting to that for crystallizing can be performed quite rapidly. The need for having the sample come to the desired crystallizing temperature rapidly is that, with large undercoolings, no appreciable crystallization occurs prior to thermal equilibrium being attained. Unlike the method employing the constant temperature bath, when using the microscope, and a temperature-controlled stage, only a limited number of spherulites are observed, but each is observed during the entire time of crystallizing, i.e., the field is not changed. The measurement of the dimensions of such spherulites can be made visually employing a bifilar eyepiece, but it is usual to record the growth sequences on film. Again, a camera back using some type of Polaroid@film can be employed, but this is also a manual operation. It is therefore customary to use some kind of automated camera. Many researchers have found that a 16 mm camera-microscope system is best. In this system an intervalometer is used to advance the film and operate the shutter sequencing. The use of photographic techniques does impose some limitations on the experiment. One wishes to take many pictures and to have each exposure time as short as possible. This implies the use of a high-intensity light source, usually having associated with it a large amount of infrared illumination. To avoid heating of the sample by such radiation, the sample should only be illuminated during exposure of the film. This means that either some type of flash illumination should be employed or that an auxiliary shutter

10.3.

MEASURING CRYSTALLIZATION RATES

36 1

that only allows illumination of the sample during the film exposure be used. A xenon flash controlled by the intervalometer is ideal for this purpose. Heat filters should still be used. A high-speed film having small grain structure is advisable. As in any good cinephotomicrographic setup, the camera-microscope equipment should be protected from the adverse effects of vibration. In general the observations are done using a polarizing microscope. With crossed Nicol prisms (film-type elements deteriorate under intense illumination) the crystal appears brightly (due to birefringence) against the otherwise black (liquid) background. For the purposes of accurate spherulitic measurement it is more important to have the boundary of the crystal properly defined than to have internal spherulitic structural details visible on the film. For this purpose a high-resolution, high-contrast film has been found to be most satisfactory. The resulting photographs produced from such film show the spherulites’ boundary in sharp relief against a black background. In each isothermal growth experiment at least ten pictures taken at fixed intervals during the experiment should be used. Any deviation from linearity in the resulting growth vs. time plot should be viewed with suspicion. In the case of linear polyethylene, it was noted that the thin films, providing one did not use the growth near the periphery of the film, showed little or no effect of thermal degradation even when heated at crystallizing temperatures for several hundred hours. However, the experimenter must always be on the lookout for such effects. With either the in situ measurements or with the constant-temperature bath experiments, it is wise to repeat the crystallization studies at one temperature frequently in order to detect degradation in the sample. By this technique any change in growth characteristics of the sample can be easily noted. If degradation occurs it is often necessary to use a new set of sample preparations for each growth experiment. 10.3.2.2.4. ANALYSIS OF D A T A . No attempt has been made in this part to rigorously drive the equations relating to the kinetics of the crystallization of chain-folded polymers nor to examine critically the assumptions used in deriving the working relationships. For these details the reader is referred to the excellent review on the subject by Hoffman et ~ 1 . ~ The working relationship used in the analysis of growth rates obtained by any of the methods described in this section is Eq. (10.2.2) with the F*/kT factor replaced by U * / R ( T - T,) and a temperature correctionf to the driving force, as follows: G = Go exp[- U * / R ( T - T,)] exp[-K,(I/T

AT’],

(10.3.1)

where T, is taken to be the glass transition temperature minus 30°K. For

~

3 62

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NUCLEATION A N D CRYSTALLIZATION

some polymers the entire growth rate curve can be measured [e.g., isotactic p o l y ~ t y r e n enylon , ~ ~ 6,4Tpoly(tetramethy1-p-silpheylene siloxane) (TMPS),48 poly(chlorotrifluoroethylene),49 and poly(o~ypropylene)].~~ For other materials such as p ~ l y e t h y l e n e ,p~ly(oxyrnethylene),~~ ~~ and selenium,53growth rates near the maximum become so rapid that no measurements can be made and the low-temperature side of the curve has been inaccessible to date. Before any data for a given polymer can be analyzed, reasonable estimates for Tgand Tm0must be made. As mentioned previously, the experimentally observed melting points for polymers tend to be well below the value for a defect-free, extended-chain crystal both because of defects and because of the thinness of the crystal. The best methods presently available for determining Tm0are the construction of a T , vs. T , plot and extrapolating to the T , = T , line or using x-ray or electron-microscopic techniques to obtain values of I that are used in constructing a plot of T , vs. I//. Both methods are difficult and time consuming. If these types of data are not available one can “guess” at a reasonable value for Tm0by assuming that it is 6 4 ° C above the highest experimentally observed melting point for the polymer crystallized under normal conditions. A method developed in this laboratory for constructing T , vs. T, plots using differential thermal analysis is described in some detail in Section 10.3.4.3.5. Gopalan and Mandelkern” have also studied in detail the effect of molecular weight and the level of crystallinity on T , vs. T , for fractions of linear polyethylene using dilatometry . Estimates for values of T , and U* can be made by a method similar to that used by Suzuki and K o v a c if~ the ~ ~ entire growth curve is accessible. In this case, the curve for G(T) rises rapidly as the temperature is lowered, passes through a maximum, and then rapidly drops as the temperature is lowered even further (Fig. 4). The rise is controlled by the factor exp[ - ( K g / T ATf)and the fall at temperatures below the G ( T )maximum is largely controllkd by the factor exp[-(U*/R(T - Tm)].It is for this case that the values for U* and T can be determined with some confidence. T. Suzuki and A. J. Kovacs, Polym. J . 1, 82 (1970). J. H. Magill, Polymer 3, 655 (1962). 48 J. H. Magill, J . Polym. Sci.,Part A 27, 1187 (1969); ibid. 25, 89 (1967);J . Appl. Phys. 35, 3249 (1964). 4 9 J. D. Hoffman and J . J. Weeks, J . Chem. Phys. 37, 1723 (1962). J . H. Magill, Makromol. Chem. 86, 283 (1965). 51 J. D. Hoffman, G. S. Ross, L. J. Frolen, and J . I. Lauritzen, Jr., J . Res. Natl. Bur. Stand. 79, 671 (1975). 52 E. Baer and D. R. Carter, J . Appl. Phys. 35, 1895 (1964). R. G. Crystal, J . Polym. Sci.,Part A-2 8, 2153 (1970). M. Gopalan and L. Mandelkern, J . Phys. Chem. 71, 3833 (1967). 48 47

10.3.

36 3

MEASURING CRYSTALLIZATION RATES -3

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Briefly, the technique used, once the value of Tm0is determined, is to obtain the best fit of a linear plot of In G + U * / R ( T - T,) vs. 1/T AT’by varying values for U* and T,, where G is the measured linear growth rate. Hoffman et al.“ found that for a fairly large number of polymers, values of U* in the vicinity of 1000-1600 cal/mole and T, values about 30°K below Tg give excellent fits. Even in those cases where the entire growth curve is not accessible, a reasonable analysis can be carried out. At temperatures near Tmo, the slope K , is relatively insensitive to the selection of values for U* and T and values of 1500 cal/mole and Tg - 30”K, respectively, appear to be reasonable choices for most of the materials studied to date. Due to the

3 64

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NUCLEATION A N D CRYSTALLIZATION

very long extrapolations involved under these circumstances, one should not put too much reliance on the values for Go obtained. In the case of dilute solutions, Eq. (10.2.2) can be used to analyze the growth rate data in the same way as for crystallization from the melt. One simply replaces the factor U * / R ( T - T,) by AH*/RT, where AH* is the activation energy of diffusion in solution (= 1-3 kcal/mole), and replaces Tmoby the dissolution temperature T D o . TDois obtained by plotting the dissolution temperature vs. 1/1 and extrapolating to an infinite I value, i.e., 1 / 1 = 0.55 When dilute solution data are plotted in this way, the slopes of the lines obtained are nearly the same as those obtained for bulk samples, indicating that the fold surfaces that are obtained in both types of crystallization are thermodynamically similar. Once a value for G has been determined experimentally by any method such as optical microscopy, light scattering, or dilatometry (either in solution or bulk), the manner for obtaining u and ceis similar. One plots In G + U * / R ( T - T,) vs. 1/T ATf (In G + AH*/RT in the case of solutions) and determines the slope K , and the intercept Goby a least-squares fitting procedure. L a ~ r i t z e nand ~ ~'Hoffman and c o - w ~ r k e r shave ~ ~ shown that the value for K , (for bulk crystallization) depends upon whether growth occurs by a single nucleation act followed by rapid completion of the growth strip or by multiple nucleation acts followed by slow spreading. This behavior has been termed regime I and regime I1 growth, respectively. For regime I, K , = 4b0crueTmo/Ahfk,and for regime 11, K , = 2 b o a a eTmo/Ahfk. Here as before, bo is the monomolecular layer thickness, Ahf the heat of fusion, k the Boltzmann factor, and cuethe product of the lateral and end surface free energies. L a u r i t ~ e nhas ~ ~ devised a test to determine which type of nucleation occurs in a given polymer. The reader is referred to this paper for the details of the test. Polyoxymethylene and polypropylene conform to regime I, Poly(chlorotrifluoroethy1ene) has been shown to conform to regime I, at least ir. the nucleation-controlled growth region (i.e., at temperatures near the melting point) whereas isotactic polystyrene follows regime I1 type behavior. In polyethylene both types of behavior can be observed in the molecular-weight region from about 20,000 to 100,000. For a molecular weight of around 30,600 the transition occurs sharply with an attendant change of morphology (see Fig. 5 ) . Above 127°C axialites are formed and regime I1 type kinetics are observed. Maxwell and Mandelkern5' have observed these same effects in polyethylene fractions. Huseby and H. E. Bair, J . Appl. Phys. 39, 4969 (1968). J. I. Lauritzen, Jr., J . Appl. Phys. 44, 4353 (1973). 57 J. Maxwell and L. Mandelkern, Macromolecules 10, 1141 (1977). ss T. W.

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MEASURING CRYSTALLIZATION RATES

365

FIG.5. Transition in growth behavior of a sample of polyethylene, M, = 30,600 as a function of the crystallization temperature (see also Fig. l).44

However, in their fractions the shift of regimes was more gradual as was the change in morphologies. Up to the present time no other polymer has been found that shows both behaviors in a single sample. However, the experimenter should be aware of the possibility and not just assume that measurements at two extreme temperatures will necessarily define the growth curve. Once the regime is established, it is a trivial matter to determine the product cuefrom K , . 10.3.2.3. The Intensity of Depolarized Light as a Measure of Crystallinity. When the average size of the spherulites is too small to be measured directly, or when the crystallization rate is too large to allow camera-microscope techniques to be used, it is still possible to modify the apparatus so as to obtain a great deal of useful information. A photomultiplier tube is substituted for the microscope eyepiece and/or cinephotographic equipment and the intensity of the depolarized light is measured or recorded as a function of time or of temperature. The technique is not only useful in determining isothermal growth rates, but also can be used to determine experimental melting points or solidsolid transition temperatures.

366

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The experimental details are similar to those described previously using the cinephotographic equipment. The only major difference lies in the need to first calibrate the photomultiplier. For this, a series of calibrated neutral filters is used, measuring the transmitted intensity with a fixed source and only the polarizer in place. Once linearity is established, or departures from linearity known, the growth experiment is conducted in exactly the same fashion as if a camera were being used. The resulting sigmoidal plot of intensity vs. time provides in itself much useful information regarding the crystallization process. For a quantitative interpretation of the data, the intensity function is normalized. The Avrami equation (10.2.1.) is used, where (1 - x) is defined as ( I , - Zt)/(Zc - lo). I, is the intensity at any time t , I , the initial intensity of the depolarized light when the sample is liquid, and I, the intensity of the light scattered from the crystallized sample. The normalization cornpensates for both changes in illumination intensity and changes in sample thickness as different samples are employed. The above definition of 1 - x implies that the intensity of the depolarized light is directly proportional to the quantity of crystalline material present. Such an assumption is reasonably well satisfied if the final spherulitic size is less than the thickness of the sample, a condition that requires a profuse spherulitic growth. The Avrami integer n is the slope of the plot of In[ - In( 1 - x)] vs. In t. The rate constant K is the slope of the plot of [-In( 1 - x)] vs. tn. The rate constant can also be calculated from the half-time of crystallization tl,z,using the relationship K = In 2/(t1,#. This technique has been used in investigating the growth kinetics of many polymers including nylon 6,5* nylon 6,6,5g poly(3-methyl butene~-~~ poly(Cmethy1 pentene- 1),61 isotactic p o l y ( p r ~ p y l e n e ) , ~and poly(ethy1ene t e r e ~ h t h a l a t e ) . ~ ~ 10.3.2.4. The Measurement of Crystal Growth by Means of Light Scattering. The technique of measuring the growth of spherulites by correlation with intensity curves from light scattering is a very powerful one. The polarization direction is quite important; different information is obtainable depending on whether the scattered light is observed using the J. H. Magill, Polymer 3, 43 and 655 (1962). J. H. Magill, Polymer 2, 221 (1961). 6o 1. Kirshenbaum, R. B. Isaacson, and W. C. Feist, J . Polym. Sci., Part B 2, 897 (1964). 1. Kirshenbaum, W. C. Feist, and R. B. Isaacson, J . Appl. Polym. Sci. 9, 3023 (1965). J . H. Magill, Polymer 3, 35 (1962). J. H. Magill, Nature (London) 191, 1092 (1961). C. W. Hock and J. F. Arbogast, Anal. G e m . 33,462 (1961). 65 K. G . Mayhan, W. J. James, and W. Bosch, J . Appl. Polym. Sci. 9, 3605 (1965). 58 58

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H, arrangement (incident light is vertically polarized, whereas the analyzer is horizontal) or by using the V, configuration (both polarizer and analyzer vertical). The apparatus is basically very simple. A light source is polarized. The polarized beam travels through a thin-film sample in which scattering occurs. This scattered light travels through the analyzer and onto a screen perpendicular to the optic axis. The spherulitic radius is defined by the following relationship:

R

=

U h [ h sin(em/2)],

(10.3.2)

where is the average spherulitic radius, A the wavelength of the light source, emthe scattering angle in the polymer sample, and U a form factor having a value close to 4. When the H, mode of analyzer-polarizer is used, the observed scattering pattern is that of a cloverleaf with four lobes. 8, is the scattering angle corresponding to the point of maximum light intensity in the lobes. The light source should be of high intensity; mercury lamps and laser sources are perhaps the most common. Either high-speed photography or a mobile photocell scanner may be used. The scanning photocell moves back and forth between two opposite leaves of the cloverleaf. With the scanning device it is also possible to have a cell permanently fixed at the optic axis, which continually records the intensity of the depolarized light. Thus, a plot of intensity vs. time can be used to determine the overall crystal growth as was described previously in Section 10.3.2.3, while at the same time determining the radial growth of the average spherulite. The use of wide apertures allows the measurement to be averaged over many spherulites; conversely, very small apertures or slits may allow observations to be made on sections of single spherulites. When the light-scattering technique is used as well as measurements using the optical microscope, excellent agreement exists between the two methods, particularly in the range of spherulitic size from 1-50 pm. The method is particularly valuable when the profuse growth of spherulites prevents the attainment of the size necessary for the observations using the optical microscope. In practice, the apparatus usually has the thin-film sample mounted in a thermostated environment that includes an additional compartment for temperature preconditioning. This separate compartment may be a hot oven used for melting the sample or may be double-compartmented, having one compartment for melting and the other for quick quenching in order to produce a glass. The apparatus is very useful in observing the entire growth curve, which in some instances is experimentally accessible for over 100". In

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those instances where a glass can be formed, the growth can be studied when the sample is warmed from the glassy state as well as when it is cooled from the liquid state. At any given growth temperature the resulting growth rates may be different depending on whether the growth temperature was approached from a molten or a glassy sample. There are many references to the use of the light-scattering technique to the problems of crystal growth in polymers. For further information on the development of theories and applications of the method, the reader is referred to Stein et ul.sa-72 10.3.2.5. Infrared Absorption Technique. Infrared studies can be carried out on any good spectrophotometer using prisms of sodium chloride, potassium bromide, calcium fluoride, etc. The usual technique is to place a film of polymer along with a thermocouple between sodium chloride or other suitable plates and to enclose the specimen in a heating cell mounted on the spectrophotometer. (The sample can also be ground up and a potassium bromide disk containing approximately 1% polymer can be prepared. The disk can be placed in a temperature-controlled environment and the spectrum scanned at different temperatures. Care must be taken in the grinding process since it is known that this process can lead to changes in the polymersample.) Using an instrument that has both preand postsample chopping of the beam to be certain that radiation from the sample at elevated temperatures does not lead to errors in the absorbance measurements is also recommended. Initially, the spectrum is run as a function of temperature, the sample being cooled from above its melting point to room temperature. Generally, bands can be assigned to crystal and amorphous regions. Once these bands are assigned the intensity of each band or bands can be measured as a function of temperature and time. The intensity D is defined as D = log(Zo/Z), where Zo is the height of the background trace and Z the height of the band peak. The thickness of the sample, of course, must be constant. If one defines 0,as the measured intensity of the crystalline sample, D, as the intensity for the melt or amorphous region, and D t as the intensity of the band at time t, the Avrami equation becomes 1 - y, = (D,- D t ) / ( D ,- 0,). Frequently, a ratio is taken between the intensities of the crystalline and amorphous bands. This procedure elimiA. Plaza and R. S. Stein, J . Polym. Sci. 40, 267 (1959). R. S. Stein and M. B . Rhodes, J . Appl. Phys. 31, N o . 1 1 , 1983 (1960). Ba R. S. Stein and P. R. Wilson, J . Appl. Phys. 33, 1914 (1972). V. G . Baranov, A. V. Kenarov, and T. I. Volkov, J . Po/ym. Sci., Purr C 30,271 (1970). F. van Antwerpen and D. W. van Krevelen, J . Polym. Sci., Purr A-2 10,2409 (1972). 71 F. van Antwerpen and D. W. van Krevelen, J . Polym. Sci., Purt A-2 10, 2423 (1972). 72 R. S. Stein and A. Misra, J . Polym. Sci. 11, 109 (1973). O7

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369

nates the problem of variations in the thickness of the samples. The crystallization rate constant can be calculated from the half-time of crystallization fIl2 using the usual relationship K , = In 2 / ( 1 ~ / ~ The ) ~ . K , values are then analyzed in the usual way. Several studies comparing the results of an Avrami-type analysis on data obtained from density m e a s u r e m e n t ~dilat~metry,’~ ,~~ and x-ray diffraction showed essential agreement between the results of these measurements and those obtained by careful infrared techniques. 10.3.2.6. The Use of Dielectric Measurements for Determining Crystallization Rates.? In 1963 it was that the measurement of the relative permittivity (dielectric constant) as a function of the fraction melted for a pure material was a very effective method for measuring purity in such systems. For most polymers there exists a linear relationship between the relative permittivity and the fraction of polymer crystallized. In general, the relative permittivity has its maximum value for the amorphous phase and its minimum value for the crystalline phase. It is therefore possible to measure the development of crystallinity in a sample at different temperatures either by measuring the change of the relative permittivity or the capacity of a capacitor filled with sample as a function of both time and temperature. The resulting data are analyzed using the usual Avrami relationship, Eq. (10.2.1.). In this case 1 - x = (C, - C , ) / ( C , - Ca),where Ct is the relative permittivity at time t , C , the relative permittivity for the pure crystalline phase, and C , the relative permittivity for the melted sample. The value of 1 - x is related to the growth rate constant K in the usual way. An application of this technique to a study of the crystallization of Neoprene is described in a paper by Simek and M i i l l e ~ - . ~ ~ 10.3.3. Methods Using a Bulk-Type Specimen 10.3.3.1. Sample Preparation. Generally, “as-received” samples are not suitable for dilatometric or other related techniques that require a bulk-type sample. The method required to clean the sample is highly dependent on the material to be studied but generally requires dissolution in a suitable solvent, filtration, precipitation, and drying under vacuum T. Okada and L. Mandelkern, J . Polym. Sci., Purr A-2 5, 239 (1967). M. Hatano and S. Kambara, Polymer 2, 1 (1961). 75 C. Baker, W. F. Maddams, and J. E. Preedy, J . Polym. Sci., Polym. Phys. Ed. 15, 1041 (1977). 76 G . S. Ross and L. J . Frolen, J . Res. Nntl. Bur. Stand., Sect. A 67, 607 (1963). 77 I. Simek and F. H. Muller, Kolloid-Z. & 2. Polym. 234, 1092 (1968). 73 74

t See also Volume 6B, Chapter 7.1.

3 70

10. NUCLEATION A N D CRYSTALLIZATION

conditions to remove the solvent. Frequently, at the high temperatures needed to dissolve the samples, oxidative degradation occurs and one must always check with GPC,viscosity, or other similar measurements to be certain that the sample has not been adversely affected during the process. Addition of a suitable antioxidant or operation under vacuum or in an inert atmosphere is required if degradation is likely to occur. As mentioned previously, the crystallization usually proceeds via a heterogeneous nucleation step. The number of heterogeneities can be controlled to some extent. Heating the sample in the apparatus to a temperature well above the melting point destroys some of these nuclei. The cleaning and filtering techniques described in Section 10.3.2.1 are also effective in reducing the number of active nuclei. Reducing the number of active sites produces fewer crystals in the sample and consequently allows one to follow the crystallization process for a longer period of time prior to the impingement of adjacent crystals. However, particularly at crystallization temperatures near the melting point, the process becomes very slow and one often wishes to have many crystallization sites in order to shorten the period needed for observation. This can be accomplished using the self-seeding technique described in Section 10.3.5.1. Once the pretreatment is completed the sample must be prepared in a suitable shape and size for use. In the case of dilatometry the sizes of the sample can be varied from as small as 50 mg to as large as 100 g depending on the design of the dilatometer, i.e., the size of the sample bulb and the length and diameter of the capillary tube, the thermal properties of the confining liquid, and the thermal properties of the polymer itself. Once the size of the sample has been set, the sample is generally molded at a suitable temperature into the desired shape, preferably under vacuum to assure that no voids are contained in the sample. Sheets of rolled material can be cut into strips, polymer can be extruded in the form of strings, or samples can be molded into various shapes and sizes using commercially available heated presses. 10.3.3.2. Dilatometric Techniques. Volume dilatometers have been found to be very useful in studying the development of crystallinity in a large number of polymer systems. The method is relatively simple and the volume changes are a very sensitive measure of changes in crystallinity. It has the further advantage that the temperatures can be controlled for the long periods of time required when polymers are crystallized at temperatures near their melting points. The apparatus and procedure most generally used in performing dilatometric experiments are similar to the ones developed by BekkedahP in his early studies on the crystallization of natural rubber. The published N. Bekkedahl, J . Res. Natl. Bur. Stand. 42,

145 (1949).

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MEASURING CRYSTALLIZATION RATES

37 I

work on the use of dilatometry generally tends to emphasize the results obtained by the method rather than a detailed description of the method. Bekkedahl's paper is highly recommended to anyone contemplating the use of dilatometry in that it describes construction of the dilatometer, calibration techniques, the method for adding the sample, as well as the choice of and the method for adding the confining liquid. In addition, the experimental technique is presented including a detailed procedure for performing the necessary calculations (well illustrated using the data he obtained for butyl rubber). A detailed discussion of errors involved as well as the overall precision of the method is also included. Once the dilatometer is loaded with the polymer specimen and a confining liquid such as mercury, the dilatometer is heated (usually in an oil bath) to a temperature TI, which is 10-20" above the melting point for the period of time necessary to be certain that all of the sample has melted. The dilatometer is then placed in a second thermostated bath, which is set at the desired crystallization temperature T , . Once thermal equilibrium is established, the height of the mercury in the capillary is read with a good cathetometer as a function of time. Initially as the polymer crystallizes, the mercury level drops fairly rapidly (state I, primary crystallization) but, as the polymer approaches its maximum level of crystallinity, the mercury level changes very slowly and for a long period of time (stage 11, secondary crystallization). The specific volume of the polymer can be calculated from the changes in height of the mercury. Once the values f o r v are determined, x,the degree of crystallinity, can be calculated for each measured point using the relationship 1-x = Herev, is the specific volume of the supercooled amorphous phase,vc the specific volume of the crystal (usually determined as a function of temperature from X-ray data), a n d v , the specific volume of the polymer at any time t (usually obtained from density measurements). In the bulk crystallization of polymer systems the Avrami relationship holds reasonably well only for the initial stage of the crystallization (stage I) prior to the time when the crystals impinge, as long as no new nuclei are formed as a function of time. As mentioned previously, the Avrami n depends on the type of nucleation that occurs as well as the shape of the crystalline entity formed. For this reason, it is recommended that a thin-film specimen of the polymer be examined with an optical microscope in the. desired range of crystallization temperatures to find out whether homogeneous or heterogeneous nucleation occurs, as well as to gain information concerning the crystal forms that may be present at different temperatures. For spherulitic growth, where the growth rate G is constant with time

v

(v, vc)/(v,v,).

372

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NUCLEATION A N D CRYSTALLIZATION

and the Avrami n is 3 when the nucleation is heterogeneous, the relationship x = xw[l - exp(-Kt3)] describes each sigmoidal-shaped isotherm and the isotherms are superimposable. Here xw is the value of x at the onset of stage I1 crystallization; K is the bulk crystallization rate constant and is defined as 4rv,,G3/3, where yo is the number of nucleation centers per cm3 at the beginning of the experiment. G can be determined from a set of experiments at several temperatures if vo is known or if vo does not change as a function of temperature. If one then plots log 4rvoG3/3 + U*/2.303R(T - T,) vs. 1/T AT, the slope is equal to -4bocr~eT,0/2.303 Ahfk and the value for C T C T ~can be calculated (see Section 10.3.2.2.4). The dilatometric technique has been applied to determine the bulk crystallization rates for a very large number of polymers. Wunderlich has collected information on many of these systems along with the appropriate references in Table VI.8, p. 216, in his recent book.4 Mandelkern’O has examined in detail the application of the Avrami equation to partially crystalline polymers. Kovacs and MansonaOhave used a differential-type system to study crystallization from solution. MagilP has studied a polymer system that contains two or more types of spherulites at the same time, each with its own nucleation and growth kinetics. Doll and Lando,a2Armeniades and B a e ~ and ,~~ Maeda and Kanetsuna,fflto mention a few, give examples of the use of high-pressure dilatometry in studies of the effect of pressure on bulk crystallization and melting behavior. 10.3.3.3. The Density Balance Technique. In this method a fairly large sample (approximately 4 g) is melted while surrounded by an inert contact liquid, and then transferred to a second bath, which is maintained at the crystallization temperature. The weight of the sample in liquid is measured continuously as crystallization proceeds by means of a suitable balance. The density of the sample is calculated at any time from the equation

+

d = dLWO/(WO - W ) ,

(10.3.3)

where d Lis the density of the contact liquid, Wothe weight of the polymer sample in V C I C U O , and W its weight in the oil. L. Mandelkem, “Crystallization of Polymers,” Chapter 8. McGraw-Hill, New York, 1964. A. J. Kovacs and J. A. Manson, Kolloid-2. & Z . Polym. 214, 1 (1966). J. H. Magill, J . Polym. Sci.. Part A 4, 243 (1966). 82 W. W. Doll and J . B. Lando, J . Macromol. Sci.. Phys. 2, 219 (1968). C. D. Armeniades and E. Baer, J . Macromol. Sci., Phys. 1, 309 (1967). Y. Maeda and H. Kanetsuna, J . Polym. Sci., Polym. Phys. Ed. 13, 637 (1975).

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373

The experimental technique most commonly used is as described by Keller, Lester, and Morgane5 in their study on poly(ethy1ene terephthalate). The contact liquid used in their experiments was silicone oil. A container that consisted of two concentric. cylinders mounted on a common base was constructed from fine-mesh stainless steel gauze. This fine gauze was not only a suitable container for the molten polymer, but ensured good contact between the surface of the polymer under study and the contact liquid, making it possible to obtain an accurate record of the change in weight of the polymer as crystallization proceeds. Since polymers frequently react with moisture when they are fused it is usually important to remove all traces of moisture. This is done by heating the polymer, preferably in a powdered form, to an elevated temperature (below the melting point) under vacuum for several hours. After drying, the polymer is placed in the stainless steel container and again evacuated to remove all entrained air. The gauze container is then placed in a closely fitting, easily detachable glass tube filled with silicone oil or other suitable contact liquid while still under vacuum. Two silicone oil baths are placed under the balance, one for melting the sample and the other for crystallizing it. A convenient method for maintaining the oil at a constant temperature is by using vapor baths. Any temperture can be maintained by boiling a suitable liquid in an outer jacket surrounding the oil. The pressure on the boiling liquid is adjusted and maintained with suitable manostates and the temperature can be easily controlled to within +O.l"C. The sample, dried and immersed in silicone oil as described above, is immersed in the melting bath for approximately 15 minutes, and then quickly transferred to the second bath, which is maintained at the preselected crystallizing temperature; the glass tube is detached and allowed to sink to the bottom of the boil bath, and the gauze container and the polymer sample are suspended from the arm of the balance. Thermal equilibrium is generally attained within 10 minutes. After equilibrium is reached, the apparent weight of the sample is measured at suitable time intervals (2 to 3 minutes) until it becomes constant. Weighings should be to the nearest milligram. A semiautomatic balancing device, incorporating a photocell that has been described by Allen and Wrighte6can be used to greatly reduce the tedium of these measurements. When the experiments are complete, density-time curves for each experimental crystallization temperature can be constructed by plotting the fraction phase change 1 - x against time. Here xI is the fraction of 85 A . Keller, G . R . Lester, and L. B. Morgan, Philos. Trans., R . Soc. London, Ser. A 247, l(1954). 86 P. W. Allen and R. A . Wright, J . Sci. Insirurn. 29, 235 (1952).

3 74

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remaining original amorphous phase and is calculated from the density pt at time t , and the initial and final density for the crystallization, pi and pf, from the expression 1 - x = ( pf - p t ) / (pf - p i ) . The initial density pi is taken to be the density of the molten polymer prior to the onset of crystallization and the final density pf is the density of the sample when crystallization is effectively completed. Once the density-time curves are obtained, the crystallization constant K in the expression 1 - x = exp( - K t n )can be determined. The detailed analysis of the experiments are the same as those discussed in Section 10.3.3.2. 10.3.3.4. The Density Gradient Technique. In the late 1930s, Linderstrgm-Lang and c ~ - w o r k e r s ~developed ~ - ~ ~ the density gradient method to a high degree of perfection. The usual form of the apparatus consists of two fairly large glass reservoirs connected by a vertical glass tube whose volume is relatively small. The bottom reservoir and lower half of the tube are filled with a high-density solvent or solution and the upper half of the tube is filled with a less dense solvent, which is miscible with the first material. The density at either end of the vertical tube is controlled by the density of the original liquids in the reservoirs and an essentially linear, vertical density gradient develops in 24-48 hours, which is stable for weeks and months, through the process of mutual diffusion of the liquids. By means of partial mixing or layering solutions of different densities, a stable gradient can be developed in a period of approximately 6 hours.OO The steepness of the gradient can be controlled by the choice of liquids having the proper densities and under ideal temperature conditions and in the absence of vibrations, densities can be determined accurately to six decimal places.88 The gradient tube must be carefully calibrated before use with materials of known specific gravity (i.e., aqueous zinc chloride solutions, glass floats). When the organic liquids are used in the gradient tube one must be certain that the liquids are neither adsorbed nor react with either the amorphous or the crystalline component of the polymer under study, for otherwise large errors would result. The samples used in this method are generally small pieces of quenched amorphous polymer that are crystallized at elevated temperatures and removed periodically from the crystallizing bath, washed and dried at room temperature, dropped into the density gradient column, and the density K. Linderstrgm-Lang,Nature (London) 139, 713 (1937). K. Linderstrgm-Langand H. Lanz, Jr., Mikrochim. Acra 3, 210 (1938). K.Linderstrgm-Lang,0. Jacobson, and G . Johansen, C . R. Trav. Lab. Carlsberg, Ser. Chim. 23, 17 (1938). R. F. Boyer, R. S . Spencer, and R. M. Wiley, J . Polym. Sci. 1, 249 (1946). sB

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3 75

determined. The theory and the analysis of the data obtained by this method are discussed in detail in Section 10.3.3.2. 10.3.4. Differential Thermal Analysis (DTA) as Applied to the Determination of Tm0

The difficulties associated with experimentally determined the melting point Tm0of a semicrystalline polymer have been discussed in Section 10.2.2.2.4. The procedure involves crystallizing a sample at temperature T, , and then determining its melting point T , . A plot of these T , ,T , pairs is supposed to result in a linear plot that intersects the line T, = T , at the temperature Tm0,which is the melting point of the perfect extended-chain crystal.43 Such a series of experiments can be accomplished using differential thermal analyzers. The experiments can be performed in several different ways, some of which are described below. 10.3.4.1. Crystallization Procedures. 10.3.4.1.1. COMPLETECRYSTALLIZATION. A few milligrams of powdered sample are placed in a DTA cell, melted, and then crystallized in situ at the desired T,. The thermogram is recorded. In our experiments with linear polyethylene the sample was held at the crystallizing temperature for 12 hours, the time necessary for the sample to reach its maximum crystallinity at the highest crystallizing temperature employed. The assumption was that most of the annealing or thickening would be accomplished during this 12 hour period. 10.3.4.1.2. PARTIALCRYSTALLIZATION. In this series of experiments, samples were melted and brought to the crystallization temperature. When the thermograph trace showed that from 0.5 to 1% crystallization had taken place, the melting sequence was started. In these instances the sample was held at the crystallization temperature for only the time necessary to achieve the partial crystallization. 10.3.4.2. Melting Procedures. 10.3.4.2.1. FROMCOMPLETE CRYSTALLIZATION WITH CONTINUOUS HEATING.The samples were first lowered to a temperature approximately 50" below T , . During this time appreciable crystallization takes place. The sample is then heated at a constant rate until the thermogram indicates that all melting has taken place. Figure 6 depicts such a melting sequence. 10.3.4.2.2. FROMCOMPLETECRYSTALLIZATION WITH DISCONTINUOUS OR STEPWISE HEATING.The sample may be lowered in temperature as described above, or the heating sequence can commence from the crystallization temperature T , . In either case the sample is heated very rapidly from an initial temperature TI to a temperature Tz , this temperature not being sufficiently high to melt the sample. After the thermogram

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FIG.6. Thermogram of the melting curve of a fully crystallized sample of polyethylene. The sample was heated at a constant rate.

trace has returned to the baseline, the sample is again rapidly raised to a temperature T 3 . Again, after the trace returns to the baseline, the process is continued until finally the melting is complete. The process is continued for one or two more steps until the experimenter is certain the thermogram shows only the heat capacity of the liquid (see Fig. 7). 10.3.4.2.3. MELTINGFOLLOWING PARTIAL CRYSTALLIZATION. In this technique one tries to find the temperature at which the sample can be set to just produce complete melting. This requires a series of experiments at each T, . The melting point is verified by incrementally raising the temperature afterwards and only producing the thermogram corresponding to the liquid heat capacity (see Fig. 8). Note: The technique of melting all of the crystallized sample by immediately raising the sample to its melting point may also be applied to a sample that has been completely crystallized. Figure 9 shows such a sequence using a pure indium sample. 10.3.4.3.Analysis Of Data. 10.3.4.3.1. WHEN THE SAMPLE IS COMPLETELY CRYSTALLIZED A N D THEN MELTED WITH CONTINUOUS HEATING.The heating rates used ranged from 1 to 10"per minute. Once temperatures were properly corrected for the effect of rate, there was little change in the determined melting point that could be associated with the rate of heating. As shown in Fig. 6 approximately 20% of the sample crystallized during the cooling from T , to T , - 50°C (the shaded area of the thermogram). Polymers behave as if they are impure, showing a

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377

FIG.7. Stepwise melting of a completely crystallized sample of polyethylene, first heating from T to T I producing area 1. After equilibration the temperature is raised to T 2 ,producing area 2, etc. The equilibration curves (times) have been shortened in this schematic representation. The shaded areas represent recrystallization.

I

time

-+

time

-+

FIG.8. Stepwise melting (heating) curves for partially crystallized (x 5 0.01) polyethylene. Both figures are obtained for the same sample, crystallized at the same temperature.

378

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change in melting temperature as more and more of the material is melted. The two extreme melting points that can be uniquely defined are T , , the temperature obtained by extrapolating the melting ramp back to the baseline, and Tb,the temperature corresponding to the melting temperature of the last crystals. In addition to these two extreme temperatures, it is also possible to determine the melting point of any fraction melted, this fractionfbeing defined as the ratio of the area of the thermogram representing that portion melted to the area representing the complete melt. In Fig. 6 the ratio of the area B, which includes the shaded area, against the total area A of the thermogram would be one such fraction frozen, f. In analogy with simple materials that contain a moderate amount of impurity, a plot of l/f; versus Tiis linear over llfvalues between 1.5 and 5 . If the polymer system behaved as the simple systems, T , would be that temperature corresponding to the l/f = 0 intercept. Such polymeric systems most certainly are not behaving as do the nearly pure small-molecule crystalline systems, but it is interesting that (a) the llfplot is quite linear, and (b) as will be shown later, the T, vs. T, plots are also linear. 10.3.4.3.2. DISCONTINUOUS (STEPWISE) HEATING. In this type of experiment, the value of the temperature at which the last crystals are melted is taken as T,. Figure 7 clearly shows that there is a great deal of recrystallization occurring (shaded areas) as the sample is partially melted. While it is obvious that the shaded areas (exothermic portions) represent some type of recrystallization process, the shapes of endothermic portions also suggest that there is a rather complex process going on during the partial melting sequence. Even when the melting sequence begins at the temperature of crystallization T , , the accumulative area of the thermograms, excluding the shaded areas, is still larger than that obtained during the crystallization process. However, as with melting that occurs by a continuous heating, a l/fvs. Tplot can be made. Here again, such a plot is quite linear, and a similar T , is obtained at the l/f = 0 intercept. Such a plot is reconstructed after first subtracting that portion of area due to the simple heating of the sample, i.e., the heat capacity. 10.3.4.3.3. MELTING BY IMMEDIATELY PROCEEDING TO THE T,. When samples are immediately raised to their melting point, a second peak or shoulder appears on the forward ramp of the thermogram. As shown in Fig. 9, this secondary peak is due to the heat capacity of the system. When this portion, area a is subtracted from the thermogram, the resulting area b represents the heat of fusion for the sample. The heat of fusion determined and that determined by the continuous heating method differ from each other by less than 0.5% in the case of indium. When the process is applied to polymers that have reached their maximum level of crystallinity, the resulting areas after subtracting that por-

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379

time -+ FIG.9. Melting of a pure indium sample. Starting at a temperature of T , - 1O"C, the sample was completely melted by rapidly raising the temperature to T, . The thermogram of area a alone was produced by starting at a temperature of T , - 10.2"C, and rapidly raising the temperature to T , - 0.2"C.

tion due to the heat capacity of the crystal is still larger than the area under the crystallization thermogram even when melting proceeds directly. Again this implies that the melting process is not a simple one. In the case of polymers even the shape of the melting curve changes as a function of T , . 1 0 . 3 . 4 . 3 . 4 . PARTIALLY CRYSTALLIZED SAMPLE. A series of experiments are needed to find the exact temperature at which the sample is completely melted. As stated previously, this is verified by the size and shape of the subsequently determined liquid heat capacity peaks (see Fig. 8). The melting curves are always double-peaked, again due to the heat capacity. Subtraction of this portion from the area under the thermogram gives an area identical to that found during the partial crystallization, indicating that little or no change occurs during the melting process. Figure 8A,B depict partial and complete melting on the first stepwise rise in temperature from samples crystallized to a x of 0 . 0 1 . The figures show the results of only 0 . 1 " C change in the initial temperature rise. The sensitivity of the method is clearly shown. 10.3.4.4. The T , vs. T, Plot. Figure 10 shows the results of plotting T,,, vs. T, for the various methods described. For this particular polyethylene fraction, M , = 30,600 and the polydispersity M , / M , = 1 . 1 9 . Anywhere from three to eight crystallization temperatures were used. There was an appreciable amount of scatter of the points for any particular method. Some of this work is still in progress and Fig. 10 only serves to

10.

3 80

NUCLEATION A N D CRYSTALLIZATION

FIG. 10. (1) Peak Tms vs. T , (continuous heating); (2) baseline Tms vs. T , (continuous heating); (3) l/f(stepwise heating); (4) I/f(continuous heating); (5) rapid, single-step heating from completely crystallized sample; and (6) single-step heating from partial crystallization (x < 0.01).

show the type of results that can be obtained. The temperature intercepts of the various plots and their T , - T , relationships as shown in Fig. 10 are given by: (a) From complete crystallization (1) Using a continuous heating technique (i) T , from baseline intercept:

T , = 47.457 (ii)

T,

+ O.6571Tx,

Tm0 = 138.40"C.

T , from peak melting temperatures: =

85.119

+ O.3935Tx,

Tm0 = 140.34"C.

(iii) T , from llfplot:

T , = 82.151

+ 0.4O9Tx,

Tm0= 139.00"C.

(2) T , from stepwise heating (l/f): Tm0 = 140.13"C. T , = 86.812 + 0.38O5Tx,

10.3.

MEASURING CRYSTALLIZATION RATES

38 1

(3) T, from complete crystallization, one-step melting: T, = 70.1

+ 0.501Tx,

(b) From partial crystallization T, = 52.54

Tm0= 140.48"C.

(x 5 0.01), one-step melting:

+ 0.6214Tx,

Tm0= 138.77"C.

It should be noted that using the theoretical predictions of Flory and Vrij4I with Tm0= 146.5"C,the T, value for a polyethylene sample of this molecular weight would be 144.6"C. Gopalan and MandelkernS4 in a study of the effect of crystallization temperature and molecular weight on the melting temperature of polyethylene, examined the behavior of several different fractions using dilatometry. In a sample (M,+20,000) they found extrapolated melting points of 141.3 - 0.3" and 137.7 & 0.2" for crystallinities of 0.05 and 0.8, respectively. These authors also reported a value for Tm0of 146.0° & 0.5 for a high-molecular-weight sample having a low degree of crystallinity. WeeksQoahad previously reported a value of 1453°C for a high-molecular-weight sample of polyethylene using dilatometric techniques. The particular differential thermal analyzer that was used in the experiments described herein was the Mettler TA2000.t The minimal temperature difference that can be set is 0.1"C. Consequently, none of the T,s obtained by the incremental heating can be reported with better precision than 0.1". In general, new samples were used for each melting experiment. No visible evidence of sample deterioration was observed as a result of the heating or crystallization process. Even with replicate experiments there was an unexpectedly large variation in the T,s obtained, as much as 0.5"Cbetween duplicate experiments. This is not an uncertainty that can be attributed to the apparatus and again shows the complexity of the crystallization process that occurs in even so-called simple linear polymers. The DTA itself is a very useful tool in the study of crystallization ofpolymers, but there still remains agreat deal that is not understood. 10.3.5. Crystallization from Solution 10.3.5.1. The Self-seeding Technique in Solution and in the Melt. When it is necessary to measure growth rates, either in solution or in Boa

J . J. Weeks, J . R E S .Natl. Bur. Stand., Sect. A 67, 441 (1963).

t Certain commercial equipment, instruments, or materials are identified in this part in order to adequately specify the experimental procedure. In no case does such identification imply recommendation or endorsement by the National Bureau of Standards, nor does it imply that the material or equipment identified is necessarily the best available for the purpose.

382

10. NUCLEATION A N D CRYSTALLIZATION

bulk, close to the dissolution temperature or the melting point of a sample, the experiments often must extend for weeks or months by the usual techniques. In 1966, Blundell, Keller, and Kovacsgl published a note describing the self-seeding technique in the growth of polymer crystals from solution. Briefly, they found while doing a series of dilatometer runs on dilute solutions of polyethylene and a two-block copolymer sample of poly(ethy1ene oxide-b-styrene) that the overall rate of crystallization was strongly dependent on T,, the temperature to which the suspension was heated. The rate of crystallization decreased with increasing T, up to a limiting dissolution temperature T beyond which it remained constant. Their conclusion was that there must be nucleation centers that survive the heating to T, , which speed up the subsequent crystallization, and the total number of nuclei decreases with increasing T , up to T, where they all disappear or at least reach a limiting value. Further work by Blundell and KellerQ2 showed that the nuclei consist of highmolecular-weight material that is stabilized by refolding during the period of time required to heat the sample to T,. In a later paper, Blundell and Kellerg3describe in detail the experimental technique developed for the preparation of polyethylene crystals from dilute solution that were uniform, single-layer crystals of controlled crystal habit. The development of the use of the self-seeding technique in bulk polymers was largely due to the work of Kovacs and c o - w o r k e r ~ . ~In* ~ ~ some very beautifully done studies on poly(ethy1ene oxide), it was found that growth starts simultaneously from all the persisting nuclei and gives rise to crystalline units (single crystals, hedrites, or spherulites) of identical shape and size. The concentration of the nuclei can be controlled between large limits by an appropriate thermal treatment of the sample. The experimental technique used for measuring the growth rates of poly(ethy1ene oxide) from the melt was as follows. A 5-10 mg sample contained between 'two microscope cover glasses was melted at about 80°C and subsequently crystallized at room temperature. The sample was then heated in a reproducible way to a temperature 1-2" below the melting point, where the major part of the sample melted, leaving seed nuclei that represent a volume fraction of the order of lo-* or less of the sample. The sample was then rapidly cooled to the desired crystallization temperature and allowed to crystallize for varying periods of time. The growth was stopped by quenching the sample in a dry ice-acetone D. J. Blundell, A . Keller, and A. J. Kovacs, J . Polym. Sci., Part B 4, 481 (1966).

* D. J. Blundell and A. Keller, J . Macromol. Sci., Phys. 2, 301 (1968). (L9

O5

D. J. Blundell and A. Keller, J . Macromol. Sci., Phys. 2, 337 (1968). G . Vidotto, D. Levy, and A. J. Kovacs, Kolloid-Z. & Z . Polym. 230, 1 (1969). A. J. Kovacs and A. Gonthier, Kolloid-Z. & Z . Polym. 250, 530 (1972).

10.3.

MEASURING CRYSTALLIZATION RATES

383

mixture. This resulted in a self-decorated crystal that was very easy to measure. Since all crystals are initiated at the same time by the selfseeding technique, this treatment results in a sample containing many crystals of the same size. These crystals were then measured to obtain the overall growth rate as a function of time and temperature. The data obtained are analyzed by the methods described in Section 10.3.2.2.4. It should be pointed out that when a sample is carefully cleaned by the methods described in Section 10.3.2.2.1, a very large fraction of the heterogeneous nuclei are removed. Since, as can be seen in Fig. 12, different nuclei are activated at different temperatures, this cleaning also results in a small number of nuclei that tend to become active at the same temperature. Under the conditions described for polyethylene fractions in Section 10.3.2.2.1, all crystals started to grow at essentially the same time (no induction time) and there were very few nuclei. Thus the crystals could be observed for long periods of time, i.e., until impingement with neighboring spherulites occurred. In this case the self-seeding technique offers no particular advantage. 10.3.5.2. Kinetics of Crystallization from Solution. 10.3.5.2.1. THE GENERAL CASE. At the present time, the number of studies on the growth rate of polymers from solution is very limited, primarily being restricted to polyethylene. The experimental methods used generally involve either optical microscopy, transmission electron microscopy, or solution dilatometry. Usually the polymer is dissolved in a suitable solvent, the number of growth centers set by “self-seeding,” the solution cooled to the crystallization temperature, and finally crystals are removed from the solution at set time intervals and their dimensions measured. In dilute solutions, nearly perfect single crystals are formed at temperatures near the dissolution temperature Td . As the concentration increases or the crystallization temperature decreases, the crystals formed tend to become more and more dendritic and finally approach the type of crystallization that occurs in bulk, i.e., spherulites. As discussed in Section 10.3.2.2.4,Eq. (10.2.2.) can be used to analyze the data obtained from solution. The appropriate growth rate equation is

G

=

Go exp[- A H * / R T ] exp[- K , l / T ( T d o- T , ) f l ,

(10.3.4)

where AH* is the activation energy of diffusion in solution, Tdothe dissolution temperature (from Td vs. 1 / 1 plots), and T , the crystallization temperature. In dilute solutions the growth rate is linear and constant until the solution is depleted of polymer. In solution dilatometry only primary crystallization occurs, i.e., there is no indication of secondary crystallization or

3 84

10. NUCLEATION A N D CRYSTALLIZATION

crystal perfecting. The Avrami type of equation also holds for dilute solution studies and can be applied to dilatometry results. Blundell and Kellerg6observed that in dilute solutions and in the temperature range where single crystals are formed, the growth is nucleation controlled. Concentration dependences have been studied by Seto and M ~ r i , Keller ~’ and P e d e m ~ n t eand , ~ ~Cooper and M a n l e ~ .They ~ ~ found that G is proportional to C“. The value for (Y is 0.2-0.5 for relatively low crystallization temperatures and high molecular weights and approaches 2 at temperatures near Td for low molecular weights. Furthermore, as the crystallization temperature is increased the upswing occurs at lower concentrations. Dilatometric studies at higher concentrations have been performed by Mandelkern. loo~lol He found that the Avrami isotherms were superimposable initially by a shift along the log t axis. However, the growth rate was found to decrease as more polymer crystallized. Yeh and Lambert102analyzed the systems isotactic polysytrene -atactic polystyrene and polycaprolactone -poly(vinyl chloride). They found in the polystyrene case that the linear growth rate decreased almost linearly with increasing atactic component; however, the presence of the atactic component did not affect the temperature at which the maximum growth rate occurred. 10.3.5.2.2. THEEFFECTOF STIRRING. When apolymer solution is rapidly stirred, crystallization occurs at lower undercoolings for a given concentration than in the unstirred systems, as predicted by Keller.lo3 The morphologies of the crystals produced are very interesting. A central fiber, which has been shown to be extended-chain material, forms first, followed by the formation of the familiar lamellar chain-folded crystals perpendicular to the central fiber and spaced at fairly regular intervals (200-1000 A). This type of morphology has been called shish kabob by Pennings et al. lo4-lo6 D. J. Blundell and A . Keller, J . Polym. Sci., Part B 6, 433 (1968). T. Set0 and N. Mori, R e p . Prog. Polym. Phys. Part Jpn. 12, 157 (1969). ga A. Keller and E. Pedemonte, J . Crysf. Growth 18, 111 (1973). M. Cooper and R. St. J. Manley, J . Polym. Sci., Polym. Lett. Ed. 11, 363 (1973). L. Mandelkem, J . Appl. Phys. 26, 340 (1955). lol L. Mandelkem, Polymer 5, 637 (1964). lo’ G . S. Y. Yeh and S. L. Lambert, J . Polym. Sci., Part A-2 10, 1183 (1972). lO3 A. Keller, J . Polym. Sci. 17, 291 (1955). Iw A. J. Pennings, in “Crystal Growth” (H. S. Peiser, ed.), p. 389. Pergamon, Oxford, 1967. loS A. J. Pennings and A. M. Kiel, Kolloid-Z. & Z . Polym. 205, 160 (1965). m A. J. Pennings and M. F. J. Pijpers, Macromolecules 3, 261 (1970). ga

10.4.

NUCLEATION

385

The shish kabob-type growth is also observed when a bulk sample is crystallized under stress such as that induced by stretching. This process is called row nucleation and the number of crystals that develop perpendicular to the applied stress increases as a function of stress.lo7 It is not within our scope here to go into all the details concerning crystallization under stress. For further discussion of experimental techniques and theories regarding this interesting and important type of crystal growth, the reader is referred to the recent book by Magill'O* and the references contained therein. 10.3.5.2.3. CRYSTALLIZATION UNDER PRESSURE. W u n d e r l i ~ hhas ~~~ shown that when a polymer solution is crystallized under different pressures, the usual chain-folded type of crystal is produced and there are no changes in crystal thickness at constant undercoolings. The primary effect of hydrostatic pressure is a slight change (increase) in dissolution temperature and a decrease in the volume of melting with increasing pressure. The effect of pressure on crystallization of a polymer from the melt is more dramatic. In the case of polyethylene, Maeda and Kanetsunallo showed that at pressures above 4000 kg/cm2, two types of extendedchain crystals were formed with different thermal stabilities. These two forms were described as ordinary and highly extended chain crystals. It is thought that extended-chain crystals are formed by a stepwise unfolding of normal chain-folded crystals. Another effect of pressure is, of course, the formation of different crystal forms. In general, for most polymers the effect of pressure is to increase the melting point and to decrease the volume of melting. For further discussion and references on pressure crystallization the reader is referred to Wunderlich's book.4

10.4. Nucleation 10.4.1. Homogeneous Nucleation

For a large number of years there has been a lively amount of interest in the precise manner in which polymers crystallize. Fundamental to the understanding of such a process is the necessity of accurately determining the values of u and ce,the lateral and end surface free energies. One method at our disposal, namely, the direct measurement of I and T , from crystals obtained at different crystallization temperatures [Eq. (10.2.1 .)I, E. H . Andrews, Proc. R . Soc. London, Ser. A 270, 232 (1962). J. H. Magill, Treatise Muter. Sci. Techno/. 10, 261 (1977). *Og B . Wunderlich, J . Polym. Sci., Part A 13, 1245 (1963). 'lo Y. Maeda and H. Kanetsuna,J. Polym. Sci., Polym. Phys. Ed. 13, 637 (1975). lop

Io8

386

10.

NUCLEATION A N D CRYSTALLIZATION

provides for a direct determination of ue. Another technique, the determination of the growth rate at different crystallization temperatures [Eq. (10.2.2.)], allows the calculation of the product uge. Finally, we have the determination of the homogeneous nucleation rate at various temperatures [Eq. (10.2.3.)], which provides a measure of the product cr2ue. Most of the work on homogeneous nucleation in polymer systems has been done on samples of linear polyethylene. The first published work on polyethylene was that of Cormia, Price, and Turnbull.lll Since then other groups112-115have reported on the homogeneous nucleation in both whole polymers and carefully prepared fractions. In all cases the investigations were carried out using small liquid droplets of the sample. The small droplets of polymer are dispersed in a suitable nonsolvent, and a small portion of this dispersion is placed in a cell on the hot stage of a polarizing microscope. When the sample is viewed through the crossed polarizer and analyzer of the microscope, only those droplets which are crystalline are seen. Against the black background the droplets appear as stars when they nucleate and immediately crystallize. The sample is first warmed to melt the droplets and then either (1) continuously cooled at a constant rate until all of the droplets are crystalline or (2) immediately cooled to some temperature within the temperature range of homogeneous nucleation. In either case a series of photographs is taken at specified time intervals. Comparison of the number of “stars” appearing in consecutive photographs thus allows the nucleation events per time and/or per degree of temperature to be calculated assuming that only a single nucleation event was responsible for each crystallized particle. Vonnegut116was the first to suggest that the existing particulate matter responsible for heterogeneous nucleation could be isolated by subdividing the bulk sample into many small pieces. If the particles were small enough, a significant portion of them would be free of foreign matter. The methods used in achieving this subdivision differ depending upon the investigators. One method is to simply grind up the sample to the desired particle size. This method was used by Turnbull and Cormia in their classic work on the homogeneous nucleation of certain n-alkanes117and R. L. Cormia, F. P. Price, and D. J. Turnbull, Chem. Phys. 37, 1333 (1962). F. Gornick, G . S. Ross, and L. J. Frolen, J . Polym. Sci., Part C 18, 79 (1967). J. D. Hoffman, J. I. Lauritzen, Jr., E. Passaglia, G. S. Ross, L. J. Frolen, and J. J. Weeks, Kolloid-Z. & Z. Polym. 231, 565 (1960). 114 J. A . Koutsky, A. G . Walton, and E. Baer,J. Appl. Phys. 38, 1831 (1967). G . S . Ross and L. J. Frolen, J . Reu. Nut/. Bur. Stand., Sect. A 79, No. 6 , 701 (1975). ll@B. J. Vonnegut, J . Colloid Sci. 3, 563 (1948). D. Turnbull and R. L. Cormia, J . Chem. Phys. 34, 820 (1961). Il2

10.4. NUCLEATION

387

subsequently employed by Uhlmann et a / . and later by Oliver and Calvertllg in homogeneous nucleation work involving a large number of normal alkanes. The other technique used was to dissolve the sample in a suitable solvent, then to subdivide the solution into small droplets, and finally to remove the solvent. Koutsky et in using this technique, achieved the initial subdivision by atomizing the hot polyethylene solution onto a suitable liquid substrate. The other investigator^^^^-^^^ used the technique first reported by Cormia et a / .l l 1 The polyethylene was dissolved in hot nitrobenzene and then slowly cooled. Upon reaching the consolute temperature, a new phase, rich in polyethylene, was formed. Upon subsequent cooling these small liquid droplets crystallized, expelling the remainder of the nitrobenzene. These micron-sized droplets were then separated from the nitrobenzene and dispersed in a suitable nonsolvent. In the cases involving the normal alkanes, the subdivision and dispersal of the particles in the nonsolvent was achieved in one step. The alkanes and the nonsolvent were mechanically blended until an average particle size of a few microns was produced. Regardless of the particular method utilized in obtaining the small droplets, it is obvious that the ultimate success of the experiment is dependent upon having as few a number of heterogeneities as possible in the final droplet population. This can only be accomplished by rigorous cleaning techniques. In the case of Ross and Frolen115the two solvents used for their nucleation work on polyethylene were xylene and nitrobenzene. In order to remove as much particulate matter as possible, filtration, distillation, crystallization, and ultracentrifugation were used. Each solvent was filtered twice through two regenerated cellulose filters, each having a porosity of 0.2 pm. Next, each solvent was distilled under reduced pressure. Only the center cut was selected. Following the distillation each solvent was cooled a few degrees below its melting point and crystallization was induced by rapid shaking. The resultant slurry of small crystals was filtered off and discarded. This type of crystal removal was performed twice. The technique is sometimes referred to as “snowing out,” the rationale being that any dirt particles present will act as heterogeneous nuclei. Then by filtering and removing the resultant crystals, the solution remaining would be dirt free. Finally, each solvent was ultracentrifuged, and only the central portion was kept. It was later found that this centrifugation had little or no beneficial effect. Indeed early experiments were performed in a “dust-free’’ white bench area. This was likewise shown D. R. Uhlmann, G . Kritchevsky, R. Straff, and G. Scherer, J . Chern. Phys. 62, 4896

llE

( 1975).

M. J. Oliver and P. D. Calvert, J . Crysr. Growth 30, 343 (1975).

388

10. NUCLEATION A N D CRYSTALLIZATION

to be of little benefit. At the end of the above processes the solvents were shown to be free of particles normally "visible" using light-scattering techniques. While purity was not the main reason for the above procedures, analysis by gas chromatography indicated a chemical purity of at least 99.5 mole% for each solvent. The liquid dispersing agent used was either a silicone oil1l40r111-113*115 isooctylphenoxy-poly(ethyleneoxy1)ethanol. This chemical is a product of the General Aniline and Cilm Company under the trade name of Igepal CA-630.t The silicone oil was preheated before use to 290°C to flash off high-vapor-pressure materials, The Igepal was cleaned by vacuum distillation and filtration. A solution of polyethylene and the clean xylene was prepared and filtered through the 0.2 pm cellulose filters. In all of the filtering operations the solutions or solvents were allowed to flow through the filter network without applying any pressure. In previous experience with preparing polymer solutions prior to analysis of molecular weight using lightscattering techniques, it was found that this very slow filtering was necessary if the samples were to be free of particulate. Each solution was filtered twice. A portion of the l% by weight, polymer-xylene solution was then poured into approximately 200 ml of nitrobenzene heated to 200°C. The polymer was immediately dissolved, with the xylene rapidly boiled away. Sufficient xylene solution was used to introduce approximately 30 mg of polyethylene into the nitrobenzene. When this solution was cooled a new liquid phase, rich in polyethylene was formed. The small liquid droplets (Fig. 11A) so formed crystallized as the temperature of the nitrobenzene was further lowered. Upon cooling to room temperature the frozen droplets were separated from the nitrobenzene by filtering. Finally, a small portion of the droplets was redispersed in approximately 10 ml of clean Igepal. It has been previously demonstrated by Cormia et al."' that polyethylene was not soluble in Igepal even after standing for days at elevated temperatures. When droplets were prepared in this fashion115from the nitrobenzene solution, it was found that the droplets tended to cluster (Fig. 11B). These clusters were not broken up by ordinary mechanical agitation when dispersed in the Igepal. Ultrasonic treatment was found to be necessary in order to assume complete break up of these clusters (Fig. 11C). Koutsky and c o - w ~ r k e rused s ~ ~ microscope ~ slides having shallow concavities to hold their dispersions. In their case no cover was used and the droplets floated at orjust below the surface of the silicone. The other investigators used cells formed by fusing a 0.2 mm thick glass washer with t See footnote on bottom of p. 381.

10.4.NUCLEATION

389

FIG. 1 1 . Photographs of typical polyethylene droplet populations. (A) Electron micrograph of droplets as they are precipitated from nitrobenzene showing size distribution. (B) Droplet population dispersed in Igepal using only mechanical (shaking) agitation. ( C ) Droplet population in Igepal using ultrasonic dispersion. Note better uniformity and smaller droplet size.

an inner diameter of 0.5 cm to an 18 mm square microscope cover glass. The sample in turn was contained within the washer and a disk made from a cover glass. Gornick and co-workers112 had previously shown that cleaning of glass cells by treatment with chromic acid, followed by washing with distilled water and clean, filtered ethanol, was sufficient. The cells, including their covers, were ultrasonically scrubbed in distilled water. The resulting glass surfaces exhibited no nucleating effects upon the liquid-polymer droplets in contact. When dispersions of liquid droplets of polyethylene were cooled, it was observed that there are temperature regions where heterogeneous nucleation occurred. There were other temperatures at which no nucleation occurred. It was noted that successive preparations of droplets using the same nitrobenzene showed that a much larger percentage of the droplets nucleated at lower temperatures. Thus, it appeared that even in the nitrobenzene a snowing out process was necessary in order to effectively scavenge those particles effective in heterogeneous nucleation. The result of such consecutive droplet preparations using the same nitrobenzene are shown in Fig. 12. All of the data reported in Ross and Frolen115resulted from nucleation experiments performed on droplet populations representing the fourth precipitation from the nitrobenzene solutions. Further repetitions of the process did not seem to result in an increase of the droplet population nucleating homogeneously. When a sample is melted and then slowly cooled, there are temperature regions of relative quiescence and then other narrow temperature regions of furious activity. This is evidenced by crystallized droplets appearing

3 90

10.

NUCLEATION AND CRYSTALLIZATION

0

TEMPERATURE ("C) FIG. 12. Nucleation of polyethylene fraction (M, = 23,000) showing the effect of repeated droplet preparation using the same solvent. 1 - n/no is the fraction of droplets nu-

cleated as a function of temperature (cooling rate is O.I"C/minute). Population A is the droplets prepared from the fourth precipitation, which is assumed to be homogeneously nucleated.

as stars in the otherwise dark background of a sample being observed between the crossed analyzer and polarizer. The most primative nucleation experiment is to slowly cool the specimen, noting the lowest-temperature region at which a large amount of nucleation-crystallization is observed in the droplet population. This is assumed to be the homogeneous nucleation region. Taking the average temperature in this region, assigning a probable value for I , the nucleation frequency, and either a most probable or else a theoretically calculated value for Z,, the preexponential factor, one can then calculate by using Eq. (10.2.3.) the nucleation rate constant Ki. Thus, with a single measurement it is possible to determine a value of the free energy product uzu,. This approach was used, not with droplets, but with a thin-film specimen of polychlorotrifluoroethylene to determine the nucleation rate constant by Hoffman and Weeks.lz0 They were able to determine that J. D. Hoffman and J. J. Weeks, J . Chem. Phys. 37, No. 8, 1723 (1962).

10.4.

N UCLE AT10 N

39 1

154°C was the average temperature of homogeneous nucleation for such a bulk sample. When a molten film was rapidly cooled down, in the vicinity of 154°C it was observed that 35-40% of the crystallization occurred in approximately 5 sec at this temperature, but that no spherulites were observed by using an ordinary optical microscope. It is unusual to have polymer systems that allow this type of observation to be made, since crystal growth is normally so rapid that by the time the sample has attained the homogeneous nucleation temperature range, the sample appears completely crystalline. However, the same experiment now applied to small droplets is quite simple to do. Indeed, in their work on normal alkanes, this was the technique used by Uhlmann and coworkers,ll* although the type of treatment is considerably less satisfactory than the other two methods to be discussed. In general such experiments furnish valuable information as preliminary experiments prior to performing the more detailed ones described below. The most important assumption concerning homogeneous nucleation experiments is that indeed such nucleation can be seen and measured. In the droplet experiment it is automatically assumed that the lowest temperature region at which nucleation-crystallization is observed corresponds to homogeneous nucleation. As with studies on crystallization it is also assumed that there are no competing reactions. Consequently, an experiment similar to that described above clearly identifies the temperature region of interest. Knowing this, we can quickly bring our molten droplet population down to within a few degrees of this temperature region and then slowly cool from this point, eventually reaching the temperature at which all droplets have crystallized. If the rate of cooling is constant and if, during the course of the nucleation experiment, we have taken pictures at known time intervals, then we have the experimental data necessary to use the second method. Usually a 16 or 35 mm cinephotographic assembly is used, although a Polaroid? camera can be used. The latter is somewhat less satisfactory in that fewer pictures are taken. The film must be of a high speed, high resolution, and preferably medium contrast type, i.e., all of the crystallized droplets must appear on the film and yet the exposure time should be small in respect to the time of the experiment. Depending upon its orientation, a crystallized droplet may appear as a very bright “star” or a barely visible one. Both extremes must be clearly discernible in any photograph. Ideally the speed of the film should be such that during the required exposure, no nucleation events occur. t See footnote on bottom of p. 381.

392

10.

NUCLEATION A N D CRYSTALLIZATION

Referring to Fig. 12, we see that by the time that the temperature region of homogeneous nucleation is reached, the beginning of region A, 350% of the total droplet population has already frozen. This population is considered as a blank, to be subtracted from the total number of droplets visible at any time during the homogeneous nucleation experiment. The portion of the experiment representing the homogeneous nucleation is subdivided into at least 10 and preferably 20 or more equal time periods and pictures corresponding to these times are printed from the many frames taken of the experiment. The total number of visible droplets on each picture is determined. The difference between this number and the blank is the number of droplets that have nucleated homogeneously for that portion of the experiment. Since the cooling rate is constant, and providing we have measured T , we have data sets of T i , ti, and n , these being the temperature, the time, and the number of droplets that have homogeneously nucleated by time ti. Similar sets of data are obtained by repeating the experiment, using a different constant cooling rate. Plots of the fraction of droplets frozen vs. temperature for a series of experiments produce a series of S-shaped curves whose major difference corresponds to a shift of temperature. From such plots for identical fraction frozen lines, we can obtain AT,,,, which is the difference in temperature between two experiments. Cormia et al. (CPT) derived the following relations hip: ln(r1/r2) = 2W* ATD/kT AT,,, ,

(10.4.1)

and consequently were able to determine W * , the work necessary to form the critical nucleus, from a plot of ln(rl/r2) vs. ATD. rl and r2 are two different experimental cooling rates and ATD is the temperature difference at the midpoint of the curves produced when n/nois plotted vs. T , the undercooling corresponding to each cooling rate. In their terminology, I = K, exp[- ( W * / k T ) ] , where K, is a constant dependent upon the molecular and transport properties of the system. W* is defined as ~ T ( ~ ~ ~ , T , , , ~ / A H , ~From ( A TW ) ~* . they were able to calculate the surface free-energy product v2a,. Equation (10.2.3.) shows a I / T ( A Y ) 2 dependence, and therefore a W * from this equation would be somewhat different, but still a derivation similar to that of CPT would allow calculation of v2vefrom the plot of ln(rl/r2) vs. A T D . In such a set of experiments the difference between the smallest and largest cooling rate should be of the order of 100. By restricting the experiments to isofractions frozen of from 0.3 to 0.7, a very linear plot of ln(rl/r2) vs. AT, is obtained, and the value of v2wedetermined from this and similar treatments such as that to be described below is very reasonable. The third method111-l15 of determining homogeneous nucleation fre-

10.4.

NUCLEATION

393

quencies or, specifically, m2ue from homogeneous nucleation experiments involves measurement of such frequencies from experiments run at a constant temperature. Again, there is a series of pictures taken of the droplet population at different times. As before, one has a blank representing the nucleation-crystallization occurring prior to the homogeneous nucleation. In this type of experiment it is not necessary to run the experiment the length of time necessary to nucleate all the droplets. Instead, one can determine the total droplet population by simply dropping the temperature to a much lower one and after a brief period of time take the resulting picture; the difference in droplet count between this picture and the blank represents the total number no of droplets that could have participated in the homogeneous nucleation experiment. In describing this technique we are specifically describing that published in Ross and Frolen,l15 although the general technique is that dis~ results of preparing cussed in Cormia, Ross, and a s s o c i a t e ~ . " ' - ~ ~The the sample by ultrasonic dispersion as compared to mechanical stirring is vividly shown in Fig. 11C. The resulting uniformity of droplet size is to some degree responsible for the precision of the results described herein. In as much as the entire droplet population remains fixed in space, i.e., the droplets do not change position from experiment to experiment or during a single experiment, it is possible to divide up the two-dimensional droplet space and to compare different subpopulations. For instance, comparison of those droplets in the upper half of the field with the behavior of the droplets in the lower half of the field is useful in determining that there is a uniformity of temperature. The behavior of these subpopulations was identical, showing that within the sensitivity of the experiment, there was no discernible temperature difference. Another test of droplet uniformity or of a single rate constant is to divide up the experiment into two or three equal times. Each should give the same slope and the same precision if indeed all the droplets conform to the same statistics. The first step in the analysis of data uses the relationship

n

=

no exp(- Zvt),

(10.4.2)

where no is the total number of droplets, n the number of droplets that remain uncrystallized after any time t , v the average droplet volume. A typical plot of In n / n o vs. t is shown in Fig. 13. The data are extremely good, having correlation coefficients of 0.998 or greater. The typical experiment contained from 500 to 1000 droplets which nucleated homogeneously. By either dividing the field into three sectors or by taking the first, second, and final one-third of the droplets as they nucleated, statistics were produced that showed little deterioration from the treatment

394

10.

-4

0

NUCLEATION A N D CRYSTALLIZATION

50

100

Time (minutes) FIG.13. Typical isothermal homogeneous nucleation experiments showing the temperature dependence of the nucleation rate for a polyethylene fraction ( M , = 30,600). n/no is the fraction of droplets that remain unnucleated at any time t .

of the population as a whole. While the nucleation experiments cover only a temperature range of a few degrees, this remarkable conformity to Eq. (10.2.3.) allows considerable testing of the ensuing nucleation theories. One of the major reasons for doing the work described in Ross and Frolen115was to see if there was any molecular-weight dependence of the homogeneous nucleation rate of linear polyethylene. The actual temperature range of homogeneous nucleation was of course dependent upon the molecular weight of the sample, being directly related to AT or Tm for each sample. However, in the range of 24,000-250,000 molecular weight on the molecthere was only a very slight dependence of the product a2ae ular weight of the sample. In fact, over this range of molecular weight the data would support the contention that there was no molecular-weight dependence. To make this analysis Eq. (10.2.3.) was modified as follows: Z = l o exp(-[U*/k(T - T,)]} exp{-[30.2 (~1~~.,(k)Tm~/k(Ahf)~]} (10.4.3) X [(l + x AT)'(l + y AT)/T(AT)'.P].

10.4.

NUCLEATION

3 95

As shown by Hoffman and L a ~ r i t z e nthe , ~ ~term AF*/kT can be substituted by U * / R ( T - T , ) for a large number of polymers, including polyethylene. U* has the same value as described in Section 10.3.2.2.4. In addition, the fact that homogeneous nucleation occurs some 65" below T,,, required that some consideration be given to the dependence of u and ue on temperature. A linear d e p e n d e n ~ was e ~ ~assumed for both u and ue, the coefficients beingx and y, respectively. The quantity? is the correction term to the free energy and is equal to 2T/(T,,, + T ) . Again, this term becomes of increasing importance with the high undercoolings. When the data were fitted with the unreasonable assumption that x = y = 0 the fit was quite good, giving an average value of the surface free energy product v2we= 19,000 erg3 cm-6. From previous it had been shown that the best value for y was 0.014. In further analysis this value of y was assumed as was the theoretical value of Zo, i.e., 1X The value of x was chosen by allowing x to vary until the best correlation coefficient of fit was obtained. Again for molecular weights above 25,000 this value was found to be nearly constant, having an average value of - 0.0073. The value of the product u2uedid not appreciably change. From these results, it is our opinion that the theoretical value of Zo should be included as input data. From this work, and that where the values of u and cewere determined by other methods, a very uniform picture emerges. The consistency of such surface-free energies when determined from vastly differing techniques suggests that the underlying theories herein used are indeed sound. Unfortunately, data on a large number of polymers are not readily available. Indeed, the work of Koutsky et af.lI4 on polyethylene oxide, polyoxymethylene, nylon 6, poly(3,3-bis-chloromethyloxacyclobutane),isotactic polypropylene, and isotactic polystyrene appears to represent the sum total of homogeneous nucleation work of a definitive nature on polymers other than linear polyethylene. Even Koutsky et af. feel that their results unequivocably represent homogeneous nucleation behavior for only polyethylene and isotactic polypropylene. Experimentation involving investigation of the homogeneous nucleation behavior of other polymers, including the effect of molecular weight, is needed. There are two methods that were used with the study of normal alkanes, but that have not been employed in the study of homogeneous nucleation in polymers; namely, dilatometry 117 and differential scanning calorimetry (DSC). In particular, Oliver and C a l ~ e rwere t ~ ~able ~ to obtain the same type of data by taking the liquified-droplet dispersions and cooling at known rates in a differential scanning calorimeter. By proper treatment of the resulting thermograms they were able to obtain relative nucleation rates and hence u3that agreed with results obtained by earlier methods in-

3 96

10. NUCLEATION A N D CRYSTALLIZATION

volving droplet counting. This method could perhaps be fruitfully applied to polymeric systems. Another method attempted by the authors of this part was to cool the liquid-droplet population while listening to the noise produced by nucleation-crystallization of the individual droplets. The result of the very rapid crystallization following nucleation of an individual droplet produces sufficient noise to be detected by ultrasensitive noise-detecting systems. Preliminary work suggested that the approach was quite reasonable, although the work was not actively pursued because of the difficulty of producing a sensor (acoustic transducer) that was not adversely effected by the elevated temperatures encountered. 10.4.2. Heterogeneous Nucleation

As mentioned in Section 10.4. l . , most polymers nucleate heterogeneously at temperatures near the melting point, and different types of heterogeneities become active at different temperatures. At the present time the nature of the foreign surfaces are not understood and the mechanism of the process has not been explained, although some interesting work on the subject has been d ~ n e . ' ~ l - 'Several ~~ experimental methods have been applied to the examination of the nucleation efficiencies of foreign materials added to the polymer samples under investigation (inorganic salts, catalysts such as titanium, and other polymers). Three such methods have been the depolarization of polarized light,122-124OPtical r n i c r ~ ~ c ~and p y differential ,~~~ thermal analysis. 126;127 The effect of substrates on solution c r y ~ t a l l i z a t i o n ' ~and ~ - ~on ~ ~crystallization from the melt131-133has also been investigated. The procedure described by Chatterjee et al. lZ5is straightforward. A polarizing microscope is focused on the interface between the polymer and the freshly cut substrate. In this way the pure polymer can be observed as well as the interface. The sample and substrate are placed on a F. L. Binsbergen, Ph.D. Thesis, Groningen, The Netherlands (1969). F. L. Binsbergen, Polymer 11, 253 (1970). lZ3F. L. Binsbergen, J . Polym. Sci., Polym. Phys. Ed. 11, 117 (1973). lZ4M. Inoue, J . Polym. Sci., Part A 1, 2013 (1963). lZaA. M. Chattejee, F. P. Price, and S. Newman, J . Polym. Sci., Polym. Phys. Ed. 13, 2369, 2385, and 2391 (1975). lZeH. N. Beck and H. D. Ledbetter, J . Appl. Polym. Sci. 9, 213 (1965). lZ7H. N . Beck, J . Appl. Polym. Sci. 11, 673 (1967). lZ8E. W. Fischer, Kolloid-Z & Z . Polym. 159, 108 (1958). J . A. Koutsky, A. G. Walton, and E. Baer, J . Polym. Sci., Part A-2 4, 611 (1966). 130 J. A. Koutsky, A. G. Walton, and E. Baer, J . Polym. Sci., Part B 5, 177 (1967). 131 3. A. Koutsky, A. G. Walton, and E. Baer, J . Polym. Sci., Part B 5, 185 (1967). H. Schonhom, Macromolecules 1, 145 (1968). 133 D. R. Fitchmun and S. Newman, J . Polym. Sci., Part A-2 8, 1545 (1970). lZ1

lZ2

10.4.

NUCLEATION

397

hot stage, heated to a temperature well above the melting point of the sample, and then cooled to the desired crystallization temperature. Data on the growth rates of spherulites nucleated by the substrate and nucleated in the bulk sample are compared by direct observation, (generally photographically). Melting temperatures for the spherulites nucleated in both ways can be determined by observing the temperature at which birefringence disappears. Nucleation densities at the interface and in the bulk can also be determined. The nucleation densities are calculated using the area of the substrate interface and the volume of the bulk polymer. The larger the ratio ns/nb, where n, is the nucleation density at the substrate and n b the nucleation density in the bulk sample, the greater the nucleating power of the substrate. Using a model of folded-chain nuclei, the dependence of the rate of heterogeneous nucleation I, on the undercooling AT is described by the equation134 log I,

=

log I% - (U*/2.3kT) - [l6uue A u T , ' ~ / ~ . ~ T ( AAhr'], T)~~

(10.4.4)

where I, is an essentially temperature-independent constant, acreare the usual surface free energies for the polymer, and A a = u + uc + urn. Here u is the lateral surface free energy for the polymer, uc the substrate-crystal interfacial energy, and r, the substrate-melt interfacial energy. If the product mue is known from other measurements on the polymer (growth rates as a function of temperature), the value for A u can be obtained from the slope of the curve when log I, + U*/2.3kT is plotted vs. 1/T AT2. A u is then a measure of the nucleating power of the substrate. Other methods for preparing samples and measuring heterogeneous nucleation rates are described in the references cited in this section. The study of heterogeneous nucleation is very important and many more systematic studies are needed to explain the mechanism of the process and to predict the effect of additives on the polymer systems. Acknowledgment We wish to thank Dr. John D. Hoffman for the benefit of many helpful discussions, comments, and suggestions.

134

F. P. Price, in "Nucleation" (A. C. Zettlemoyer, ed.), Chapter 8. Dekker, New York,

1969.