Crystallization Kinetics

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Crystallization Kinetics

  1. 1. Vol. 9 CRYSTALLIZATION KINETICS 465 CRYSTALLIZATION KINETICS Introduction Crystallizable polymers form semicrystalline materials containing chains or frac- tions of chains that are trapped in nonequilibrium, amorphous states. Because of this semicrystalline nature, the crystal morphology, rather than the under- lying crystal structure, often controls the final properties of a polymer article (see SEMICRYSTALLINE POLYMERS; MORPHOLOGY). Simply by changing the processing conditions, the mechanical properties, for example, Young’s modulus, of a typical polymer such as polyethylene can be altered by several orders of magnitude. Because polymers crystallize so far from equilibrium conditions, a simple examination of the phase diagram gives us little insight into the crystal morphol- ogy that is formed or the route that is taken to its formation. To understand, and ultimately control, such behavior, it is necessary to gain an understanding of the kinetics of crystallization, as it is the kinetics of the process that define the structure and properties of the material. The above insight, gained soon after the discovery of polymer single crystals, and the subsequent discovery of chain folding (1–3), has mapped out the route taken by much of the research into polymer crystallization over the subsequent decades. It soon became clear that the thickness of polymer lamellae was controlled by the supercooling at which they were crystallized and defined by the kinetics of crystallization. The crystal thickness, or alternatively the thickness of each new crystalline layer in a growing crystal, is the one that grows the fastest (4,5) rather than the one that is at equilibrium (6). There is now a wealth of information available on the crystallization of many polymers, as well as several theories that aim to predict the crystallization rates, crystal shapes, and lamellar thickness. Crystallization kinetics is the area of polymer science that deals with the rate at which randomly ordered chains transform into highly ordered crystals, and includes every aspect of the resultant structure that is dependent on the route that was taken between those different states. It is a broad and mature area of scien- tific research, given an uncommon diversity when compared to the crystallization of small molecules because of the wide range of different chemistries and chain topologies that are available to macromolecules. These add layers of complexity that can make it difficult to find generalizing principles. For the sake of brevity, this article, therefore, concentrates on areas that are of particular interest to the author, and to principles and observations that have, in the author’s opinion, the widest applicability. As in any transformation, it is possible to measure bulk rates of polymer crystallization and to characterize the process in terms of the shape of these transformation curves. Because of limitations on space the interested reader is pointed toward the relevant literature (7–12), as this approach, although useful for quantifying experimental observations, does not attempt to provide an insight into what is happening at the molecular scale. Polymer crystallization, like many phase transitions, occurs through a process of nucleation and growth, and it is often more revealing to look at the rates of these two processes independently, and to try to gain an understanding of the factors affecting each. Although of fundamental importance, the nucleation step is still poorly understood and only Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.
  2. 2. 466 CRYSTALLIZATION KINETICS Vol. 9 occasionally studied. The next section outlines the key experimental data and the theories that have been developed to explain this data. A brief look at some more recent approaches is also given. The majority of this article considers the growth process itself, concentrating on linear, flexible polymers as this is where the most complete data set is available, but also touching on more complex materials. An outline of the theories that have been developed to explain and predict crystal- lization behavior is also given, concentrating on their fundamental assumptions and predictions rather than their mathematical intricacies. Nucleation Any supercooled or supersaturated system contains local fluctuations in order and density, which will, at times, correspond to crystallographic order. These crys- talline regions are known as embryos and offer the chance, but not the certainty, of starting the process of phase transformation. Most of these embryos are short- lived, but there is a small probability (depending on the supercooling or supersat- uration) of any particular fluctuation reaching a critical size. This critical size is the size at which it is in unstable equilibrium with the surrounding phase—it has formed a critical nucleus, and there is a high probability of it continuing to grow to macroscopic size. Nucleation can be divided into homogeneous nucleation and heterogeneous nucleation, the former being the sporadic formation of critical nuclei from the pure phase, the latter occurring at the surface of impurities within the system. In high polymers heterogeneous nucleation dominates in most cases (13). Heterogeneous nucleation can be divided into different classes; epitaxy, in which there is a lattice match [usually within 10% (14,15)] between a crystal plane in the polymer and a free surface in the “impurity”, and nonspecific surface nucleation, in which the presence of the impurity surface leads to a chain conformation closer to that in the crystal. A final class of nucleation is self-nucleation, in which during the melting of the polymer the temperature was too low, or the time at that temperature too short, to allow the whole sample to melt. This is particularly common in polymers and is caused by the broad range of melting temperatures typically present in any sample because of a combination of factors including the range of crystal sizes, molecular weight, chain perfection, and local environment. Small crystalline “seeds” are left in the sample, which then act as epitaxial sites for nucleation on cooling (16,17), leading to the name “self-seeding” being applied to this form of nucleation. Experiments on Nucleation. The most commonly observed crystal mor- phology when polymers are crystallized from the quiescent melt is the spherulite (see SEMICRYSTALLINE POLYMERS) (18). Figure 1 shows a typical polarized light mi- crograph of the growth and intersection of a spherulitic sample, from which it can be seen that the individual spherulites intersect with each other to form polygons with straight sides. It is immediately apparent that, assuming the growth rate is constant across the sample, all of the nucleation sites must have become active at the same time, that is, the growth of all the spherulites started at the same time. If the same sample is melted and then recrystallized at the same temperature, the spherulites grow back in the same place. Nucleation is heterogeneous under these most common circumstances, occurring on impurities or added nucleating
  3. 3. Vol. 9 CRYSTALLIZATION KINETICS 467 Fig. 1. A series of images showing the growth of spherulites of polyethylene. The unifor- mity of spherulite size, and the linear intersection of the spherulites once crystallization is complete, can be clearly seen. Exposures taken after the marked times. Optical micro- graphs with the sample between crossed polars. Reprinted from Ref. 19. Copyright (1982), with permission from John Wiley & Sons, Inc. agents. Measurement of the size of the spherulites gives a measure of the number of nuclei per unit volume, and the variation of this with temperature can be sim- ply obtained (20). Some polymers behave differently, nucleation occurring more sporadically, giving a measurable nucleation rate as a function of crystallizable volume at a particular temperature (21,22). An example of this behavior is shown in Figure 2 for polyethylene succinate, with the variation with molecular weight also included. Whether this nucleation is actually homogeneous, or if the number of heterogeneities and their relative activity is just low, is unclear. Great lengths have been gone to in order to obtain truly homogeneous nucle- ation rates. Initially, one of the driving factors for this work was to obtain a figure for the side surface free energy for use in secondary nucleation crystal growth theory. In systems such as polyethylene, where nucleation rates are high and pu- rity is low, it is necessary to divide the sample up into a large number of droplets
  4. 4. 468 CRYSTALLIZATION KINETICS Vol. 9 Fig. 2. A graph showing the temperature dependence of the nucleation rate for a series of different molecular weight fractions of polyethylene succinate. The molecular weight shown is the peak molecular weight as measured by GPC. mp: 1,160, 2,060, 3,380, 4,660, 6,670, 8,770, 10,980, 13,660, 18,340, 21,210. Reprinted from Ref. 23. Copyright (2002), with permission from Marcel Dekker, Inc. (19,24–27). Many of the droplets will contain impurities and nucleate relatively early in the experiment; however, some will remain amorphous for long times, and, if the rate of nucleation in these droplets is measured, the variation of nucleation rate with temperature can be obtained. If the nucleation is truly homogeneous, and the droplet size is monodisperse, the nucleation rate should, at a particular temperature, depend only on the sample volume. Recently, similar experiments have been carried out on phase-separated block copolymers, in which the crystal- lizable domains are isolated spheres in the hard (or semihard) segregation limit (28,29). An example of data obtained by these two different methods is shown in Figures 3 and 4. Nucleation Theory. Classical theories for nucleation in small molecules balance the reduction in free energy that occurs because the solid is more stable than the supercooled liquid against a surface term that accounts for the free- energy cost of creating a solid–liquid interface. For a spherical crystallite this is given by 4 3 G= π r Gυ + 4π r2 σ (1) 3 Where G is the Gibbs free energy, r is the radius of the sphere, Gυ is the difference in free energy between the bulk fluid and bulk crystalline phase, and σ is the surface free energy of the crystal fluid interface. The function G goes through a maximum at the critical nucleus size, resulting in a nucleation barrier—the maximum in the increase in free energy necessary to form the critical
  5. 5. Vol. 9 CRYSTALLIZATION KINETICS 469 Fig. 3. A composite graph showing the crystallization of micro phase-separated spheres of E/SEB63 (polyethylene block styrene–ethylene–butene). The polyethylene block exists in spherical domains with a radius of 12.7 nm in a matrix of rubbery material. At sufficiently high supercoolings the polyethylene nucleates separately in each domain. The graph shows the time course of the integrated small-angle X-ray scattering (middle curve) and wide- angle X-ray scattering (bottom curve) intensities for E/SEB63 crystallized at 67◦ C (the insets show the regions of integration). The SAXS intensity for the polyethylene (E40) homopolymer, which shows a sigmoidal time evolution, is shown for comparison (top curve, 95◦ C, similar half-time). Reprinted from Ref. 28. Copyright (2000), with permission from the American Physical Society. nucleus—given by 16π σ 3 G∗ = (2) 3( Gυ )2 For polymers the situation is changed by two factors. Firstly, the connectivity along the chain of identical crystallizing units means that the nucleus is unlikely to be spherical, and a better first approximation is a cuboid. Secondly, the end surface (the surface perpendicular to the chains) has a very different energy from the side surface. These lead to a slightly different equation, allowing for the different geometry, given by G = −υµla0 b0 Gυ + 2µυa0 b0 σe + 2σ l(µb0 + υa0 ) (3)
  6. 6. 470 CRYSTALLIZATION KINETICS Vol. 9 Fig. 4. A graph showing the rate of nucleation of droplets of polyethylene. Isolated droplets are produced by spraying from solution. As the droplets are only ∼1 µm in size, most of them do not contain large heterogeneous nucleating agents so high supercoolings can be reached. The nucleation rate can be measured directly by counting the number of droplets that have crystallized as a function of time. The data were all collected from the same preparation of droplets, which could not be cooled below 87◦ C without all the droplets crystallizing. 87.5◦ C, 88.4◦ C, 89.6◦ C, 90.1◦ , 90.8◦ C. Reprinted from Ref. 27. Copyright (2000), with permission from Kluwer Academic Publishers. Where ν and µ are the number of stems in the length and breadth of the lamella, a0 and b0 are the cross-sectional dimensions of a stem, l is the thickness of the lamella, Gν is the difference in bulk free energy between the crystalline and liquid phases, σ e is the fold surface free energy, and σ is the side surface free energy. This then leads to the rate of formation I of folded chain nuclei given by 2 − FD 32σ 2 σe Tm 0 I = I0 exp exp (4) kT kT( T)2 ( Hυ )2 which is equivalent to the well-known Turnbull–Fisher equation (30). The fac- tor F D here corresponds to the energy barrier affecting transport of material across the crystal–liquid interface, and, under the assumption that this is an ac- tivated process, leads to the first exponential term. I0 is essentially temperature- independent and depends on molecular parameters, H ν is the enthalpy of fusion 0 per unit volume, Tm is the equilibrium melting temperature, and k is the Boltz- mann constant. Heterogeneous nucleation is introduced simply by altering one of the side surface energies to allow for the relatively low energy penalty associated with the formation of a polymer-nucleating surface interface (13). Experimental studies (19,24–27) are broadly in agreement with the predicted behavior for the
  7. 7. Vol. 9 CRYSTALLIZATION KINETICS 471 supercooling dependence of nucleation rate under conditions of homogeneous nu- cleation. Equations 3 and 4 apply to nucleation with chain folding, and so for a high molecular weight polymer only one or a few molecules might be involved in a sin- gle nucleus. This situation could occur in dilute solutions, but seems unlikely as the concentration is increased, or in the melt. An alternative is to assume that the nucleation is of a bundle of stems, each stem coming from a different polymer chain, forming a ciliated or “fringed-micelle” nucleus. Now, in addition to the un- favorable entropy contribution from demixing the polymer from solution, there is an unfavorable contribution from the restrictions on the conformations accessible to the noncrystalline portion of the polymer molecules imposed by their incorpo- ration into the nucleus. Flory (31) [and later Mandelkern (32)] considered these contributions in his original work on polymer crystallization, and, although some of the initial assumptions have proven to be in disagreement with experiment, the resulting formalism is still relevant under certain situations such as the crystal- lization of parts of chains confined between lamellae. This approach leads to the following expression for the free energy of formation of a ciliated crystal, in the limit of high dilution (polymer volume fraction tends to zero): kT lnν2 G = −υµla0 b0 Gυ + 2µυa0 b0 σe − + 2σ l(µb0 + υa0 ) (5) 2a0 b0 where ν 2 is the volume fraction of polymer. Testing the above theories, which are in essence just classical nucleation the- ory applied to polymers, has proved problematic because, as already mentioned, under most circumstances nucleation is heterogeneous and often virtually instan- taneous. In the case of polyethylene, it is necessary to divide the polymer into such small volumes in order to observe something that might be homogeneous nucleation that the effect of the surfaces of the droplet might well start to play a significant role (33,34). Studies using block copolymers, although of great value, are studying nucleation in a different polymer, and the effect of connectivity to an uncrystallizable unit is complex (35). Recently, several groups have reported data collected in the early stages of crystal growth that they have interpreted as being evidence of spinodal decom- position in the melt prior to nucleation and growth (36–38). These follow from similar suggestions made in the 1980s (39–41). The interpretations are based around the observation of small-angle X-ray scattering (SAXS), usually associ- ated with the formation of lamellae, prior to the wide-angle signal that comes from crystallographic order. The evolution of the SAXS signal in this initial stage is characterized by first-order kinetics (Avrami exponent of one), and identified with a spinodal-like transformation. This is a difficult area, as identifying the point at which an increasing signal rises above the background noise is not clear- cut (38,42), and there are many problems associated with comparing wide- and small-angle data (38). It has been proposed that the components of the melt that are phase-separating are segments of the chain that are closer to, and further from, their crystalline conformation. This separation is driven by a coupling be- tween density and order in the chain, in that by aligning, for instance, similarly helical chains they are able to pack more closely and increase the local density
  8. 8. 472 CRYSTALLIZATION KINETICS Vol. 9 (43). Thus the melt separates into regions of higher order/density and lower or- der/density, and nucleation is more likely to occur in the already more ordered regions. The “phase” separation drives a change in order into a state that is still far from crystalline, but has a higher probability of crystallizing than the initially supercooled melt. If the transformation occurs under conditions of constant pres- sure, as is usually the case, it will eventually result in an overall densification of the melt. The theory is internally consistent, but there is still an active debate as to whether or not it actually applies to the experimental situation. A priori, it seems more likely that it is of relevance to polymers with more rigid chains, and crystallization at high supercoolings. Crystal Growth In this section an overview of key experimental results on the growth of polymer crystals is given, concentrating on linear molecules with simple chemistry and moderate molecular weight. Although these molecules are the most commonly used in industrial applications, there are many other types of behavior that are markedly different, and some indication of these perturbing effects is given. This will provide the data against which any theory should be tested, and it is the basis of the different theories that are discussed in the next section. Finally, over the last 10 years, there has been a spate of new ideas, due both to the data pro- vided by new experimental techniques such as synchrotron x rays and scanning probe microscopy and to the increasing power of computer simulation. These new approaches are covered in the last section. Key Experimental Results. Polymer crystals grow in the form of lamel- lae in which the chain axis is oriented approximately perpendicular to the basal plane (1). As these lamellae are typically between 5 and 50 nm in thickness, the chains must fold back on themselves, reentering the same crystal many times. The extent to which chains reenter the crystal on average at the adjacent lattice site, or elsewhere, has been a subject of argument for 40 years (44). Although the exact nature of this “chain-folding” is still not agreed upon it is this morphologi- cal characteristic that signals most clearly the importance of kinetics in polymer crystallization. It has been widely accepted since the early 1960s that the equi- librium polymer crystal contains chains that are extended. The lamellae that are commonly observed are thinner than this, as the most rapid way that the avail- able free energy can be consumed is by the formation of fast-growing thin crystals rather than slow-growing thick crystals. Figure 5 shows a plot of crystal thickness vs supercooling, showing the in- crease in crystal thickness that occurs as the melting temperature is approached (45,46). The graph includes data for both solution and melt crystallization, show- ing that it is not the absolute temperature that is the controlling factor on crystal thickness, but rather the supercooling (or supersaturation) at which crystalliza- tion occurred. Supercooling is here defined as the equilibrium melting tempera- 0 ture (Tm ) minus the crystallization temperature, where the equilibrium melting temperature is the temperature at which an equilibrium, infinite, extended chain crystal would melt. These data were collected for polyethylene, but similar data exist for a range of other polymers. The supercooling controls the crystal thickness,
  9. 9. Vol. 9 CRYSTALLIZATION KINETICS 473 Fig. 5. A graph showing the variation of the fold length (l) with supercooling ( T) for polyethylene crystallized from a variety of solvents and from the melt. In the case of solvent crystallization, supercooling is taken with respect to the so-called equilibrium dissolution temperature. For the melt-crystallized data set the equilibrium melting temperature is used. The remarkable coincidence between the curves, despite the wide range of absolute temperatures to which each supercooling corresponds, is strong evidence in favor of the kinetic origin of crystal thickness selection. Solvents: xylene, hexyl acetate, ⊕ ethyl esters, dodecanol, dodecane, octane, × tetradecanol, + hexadecane, melt crystal- lized. Reprinted from Ref. 44. Copyright (1985), with permission from Kluwer Academic Publishers. and the thickness that grows is just thicker than the minimum stable thickness at that temperature. This minimum stable thickness comes from the balancing of the increase in energy due to the presence of the crystal melt interface, with the reduction in free energy due to the bulk free energy of the crystal—that is, the Gibbs–Thomson equation. In the case of lamellae with large lateral dimensions this can be simplified to give 2σe lmin = (6) F where lmin is the minimum stable thickness, σ e is the fold-surface free energy, and F is the bulk free energy of crystallization per unit volume. Interestingly, despite this clear evidence of the influence of kinetics on mor- phology, the crystal structure itself, and in many cases even the topology of the noncrystalline fold surface, seem to be controlled by equilibrium considerations (47). Recent computer simulations (48), in which a random polyethylene fold sur- face was allowed to relax, resulted in a 201 crystallographic fold plane, in agree- ment with experimental observations of crystallization (49).
  10. 10. 474 CRYSTALLIZATION KINETICS Vol. 9 Fig. 6. A graph of the variation in growth rate with temperature for poly(hydroxybutyrate), showing the typically observed bell-shaped temperature de- pendence, with the crystallization rate reducing to zero well before the equilibrium melting temperature (∼198◦ C for this polymer) and, after passing through a maximum, again reducing to zero before the glass-transition temperature (∼0◦ C). (Unpublished data of the author.) As detailed in the article on semicrystalline polymers, polymer lamellae, when crystallized from the melt, typically form aggregates. The most common of these aggregates, if the melt is unperturbed, is the spherulite. The growth rates of spherulites have been measured as a function of crystallization temperature for a wide range of different polymers. These rates are found to be approximately constant at a particular temperature for a particular polymer sample. An exam- ple of the variation in growth rate with temperature of polyhydroxybutyrate, a biodegradable thermoplastic, is shown in Figure 6. A bell-shaped relationship is apparent, typical of glass-forming liquids. Similar data have been obtained for many other polymers, the lower limit of the growth depending on the glass- transition temperature of the material, the upper limit depending on the melting temperature, and the peak rate varying from only a few nanometers per second to hundreds of micrometers per second, depending on the polymer. The inset shows an optical micrograph of a growing spherulite from which rate measurements can be made. The rate that is measured in a typical experiment is the average growth rate of many lamellae as they grow over a distance of several microns. It is not the growth rate of an individual lamella, and is certainly not measured over length scales comparable with either the size of the unit cell, or even the thickness of a lamella. This is simply a result of using diffraction-limited optics to measure rate. It is common practice to replot ln(growth rate) against 1/T T, so as to lin- earize the data and to allow its analysis in the context of secondary nucleation theory (see section under Secondary Nucleation Theory). A large body of data
  11. 11. Vol. 9 CRYSTALLIZATION KINETICS 475 exists in which these essentially linear plots contain one or two changes in slope, and the different linear portions are associated with different “regimes” of growth (50–58). This is a difficult area, and has been the subject of debate for many years (58,60). There certainly are instances when a true change in the temperature de- pendence of growth rate occurs, in many cases accompanied by a change in the crystal morphology. A systematic survey of the effect of varying the additional transport term that is often included in the y-axis of the plot, or of the often im- precisely defined value of T m 0 , is not available. However, growth rate is clearly exponentially dependent on 1/T T at small supercoolings. The rate of growth is influenced not only by molecular architecture and tem- perature, but also by molecular weight (61). Figure 7 shows the effect of molec- ular weight on growth rate for a series of different molecular weight fractions of polyethylene succinate (62). The variation in growth rate with molecular weight leads to complex effects during polymer crystallization, as in all high polymer Fig. 7. A graph showing the variation in linear crystal growth rate with temperature for a series of different molecular weight fractions of polyethylene succinate, showing the typ- ically observed behavior. Again, the molecular weight shown is the peak molecular weight as measured by GPC. In Ref. 61 the authors obtain a “master curve” for polymer growth rate by plotting the reduced growth rate G/Gmax (where G is the growth rate and Gmax is the fastest growth rate exhibited by the particular polymer sharp fraction) against the reduced temperature T/T cmax (where T is the crystallization temperature and T cmax is the temper- ature at which the maximum growth rate occurs). This leads to the intriguing possibility of a universal relationship between growth rate and molecular weight. In particular, it is found that the molecular weight dependence of Gmax can be expressed as a power law, Gmax proportional to MWα where α depends on the adsorption mechanism for polymer molecules on the growth front and is equal to −0.5 for folded chain growth. MP: 1,130, 2,080, 3,220, 4,590, 6,570, 9,150, 10,900, 13,700, 17,900, 21,600. Reprinted from Ref. 62. Copyright (2002), with permission from Elsevier Science Ltd.
  12. 12. 476 CRYSTALLIZATION KINETICS Vol. 9 samples there is a distribution of chains with different molecular lengths. Careful studies of the molecular weight distribution within the crystallized and uncrystal- lized portions of a sample at different stages of growth have shown a remarkable degree of fractionation (63,64). At high temperatures the longest chains crystal- lize most readily and so fractionation occurs during growth, leaving the shorter molecules to either crystallize more slowly later, or on cooling if the distribution of lengths is very broad. At lower temperatures, the shorter chains are able to crys- tallize more rapidly, due to their shorter relaxation times, and thus they will have a stronger influence on the measured growth rate. The extent of fractionation, particularly apparent during solution crystallization, has lead to the suggestion that there must be a process of “molecular nucleation” (65), in which each molecule nucleates on the growth front as a separate object rather than the important step being the nucleation of parts of a chain or chains. This provides a mechanism by which the growth process can “feel” the molecular length of each chain that attaches to the growth front. A fuller discussion is given in Reference 65. Crystallization studies carried out in dilute solution have frequently been used to gain additional insights into the process of growth, as the crystallizing lamellae can be more straight forwardly examined by electron microscopy after their growth (66,67). Figure 8 shows an atomic force microscope (AFM) topogra- phy image of a typical single crystal of polyethylene, from which the very regular crystal thickness can be seen. The well-defined crystallographic shape is typical of solution-grown crystals of many (but not all) polymers, their exact shape de- pending on the underlying crystallography and the interplay of growth rates of the different crystal faces, which also varies with crystallization temperature (see SEMICRYSTALLINE POLYMERS for more detail on crystal shape). Crystal thickness is typically plotted against the equilibrium dissolution temperature, a temperature that has an ill-defined meaning (68,69), but these plots give a remarkably similar temperature dependence of thickness to that seen for melt crystallization, in the case of polyethylene. Despite technical difficulties, growth rates as a function of temperature can be measured by careful experimentation (70–75), and can be plotted to give an approximately linear relationship with 1/T T. An example is shown in Figure 9. The additional complexity that the effect of polymer concen- tration adds to the story has been studied in some detail (71–75), but is beyond the scope of this review. There are many factors that can affect the way crystallization occurs, and lead to differences from the above outlined pattern of behavior. Linear homopoly- mers consist of repeating units, all of which can crystallize. Even in this simple case, the end groups are chemically different from the rest of the chain, and this can lead to some perturbations in behavior. End groups are usually excluded to the surface of crystals (76). For low molecular weight polymers (M w < 10,000), the presence of end groups starts to play a role in the rate of crystallization (77–79). In polymers of very uniform molecular weight, such as the monodisperse alkanes (78), and sharp fractions of low molecular weight poly(ethylene oxide) (77,79), lo- cal minima in the variation in free energy with crystal thickness exist, such that the formation of crystals that are an integer fraction of the extended chain crys- tal thickness is strongly favored. In this case, steps and even minima in growth rate are observed in the region where the crystallization temperature allows thinner crystals to become stable (80). An example of these local minima in growth
  13. 13. Vol. 9 CRYSTALLIZATION KINETICS 477 Fig. 8. An AFM topographic image showing a group of single crystals of polyethylene crystallized from xylene at 70◦ C. The grey scale shows the variation in height, from which the very uniform thickness of the crystal can be clearly seen. (Unpublished data; image courtesy of Dr. A. K. Winkel). rate is shown in Figure 10 (81). These minima occur because of competition at the growth front between different metastable “phases”—for example once-folded and twice-folded crystals (82). If copolymers are crystallized, it is necessary to consider the cocrystallizabil- ity of the different chemical species (31), the length and distribution of sequences, the miscibility of the melt (ie, do the comonomers mix) (83), and many other fac- tors (84). These areas have all been extensively studied, and are still the subject of active debate. However, the recent advent of more controlled techniques for poly- merization offers hope of reaching a greater understanding. In most cases, one of the copolymer species will crystallize, excluding the other from its lattice. Crys- tals can only form if sequence lengths are sufficiently long to span the thickness of the crystal, and so an additional control on crystal thickness, on top of kinet- ics, is added. It is now necessary to sort the molecules at the growth front so as to access the crystallizable sequences, and this additional step slows the growth
  14. 14. 478 CRYSTALLIZATION KINETICS Vol. 9 Fig. 9. A graph showing the variation in growth rate with 1/T T for single crystals of polyethylene grown from different solvents. The top data set is for 0.05% Rigidex 50 in tetradecanol, the middle one of 0.05% Rigidex 50 in hexadecane, and the bottom one for 0.l% Marlex 6009 in xylene. The best fit lines (dashed) show an approximate linear dependence, while the dotted lines in the top two data sets are lines in which the ratio of the slopes to the higher temperature lines is exactly 2, as predicted by theory (see section under Secondary Nucleation Theory). Reprinted from Ref. 74. Copyright (1986), with permission from John Wiley & Sons, Inc. rate. Also, at a particular temperature, some sequences will be too short to crys- tallize (crystals of a thickness commensurate with the length of the sequence are not stable at that temperature), and so will be unable to crystallize until a lower temperature is reached. Lower total crystallinities are usually obtained, because it is not always possible for all of the sequences that are potentially crystallizable at a particular temperature to find a crystal into which to crystallize, considering the constraints due to the crystallization of other sequences in the same chain. A very similar behavior is observed with branched polymers, as longer branches
  15. 15. Vol. 9 CRYSTALLIZATION KINETICS 479 Fig. 10. A graph showing the variation in crystallization rate with temperature for the ultralong alkane C246 H494 . + extended chain growth, integer once-folded chain growth, ∗ twice-folded chain growth. A minimum in crystallization rate can be seen at the temper- ature where the transition from primary growth in different folded forms occurs. This is believed to be caused by competition at the growth front between the more stable thicker crystal and the now just stable, thinner form. Reprinted from ref. 81. Copyright (2000), with permission from Elsevier Science Ltd. are excluded from the lattice (85). The distance between branch points defines the maximum crystal thickness into which a particular chain segment can crystal- lize. If this is thinner than the minimum stable thickness for a crystal, the chain segment cannot crystallize at that temperature. Again, this leads to a slowing of the overall growth rate, a reduction in the achieved crystallinity, and a widening of the temperature over which crystallization occurs. In the case of polyethylene, which has been extensively studied because of the commercial importance of low density polyethylene, some material is believed to not crystallize until tempera- tures below −20◦ C are reached. If high pressure is added, it has been shown (86) that it is even possible for the side branches to crystallize, albeit into a hexagonal phase. As well as chain chemistry and chain length, different behaviors from the above outlined “typical behavior” can also be found because of differences in the crystallization conditions of an otherwise “typical” flexible linear polymer. Crys- tallization under flow leads to a change in morphology (87–89). Initially, oriented nuclei are formed with a largely extended chain character. These nuclei consist of the high molecular weight tail of the molecular weight distribution, both high molecular weight and polydispersity apparently being required (89–91). From these oriented backbones, plate-like lamellae then grow, largely perpendicular to the backbone, as shown in Figure 11 (92). These lamellae grow in a manner sim- ilar to that seen in the absence of flow, and have been used as model geometries for the study of lamellar growth (93). However, the growth of the backbone itself is very difficult to follow, due both to its size—although up to several microns in length, their width is typically less than 10 nm—and the very rapid rate of its formation.
  16. 16. 480 CRYSTALLIZATION KINETICS Vol. 9 Fig. 11. An AFM micrograph (phase image) showing a shish-kebab crystal of polyethylene in a matrix of molten polymer. Image collected at 135◦ C. The bright area in the top right of the image is the glass substrate. The extended chain backbone and lamellar overgrowths can be clearly seen. The scale bar represents 100 nm. Reprinted from Ref. 92. Copyright (2001), with permission from the American Chemical Society. In many commercial applications, polymers are now being used to add value through the use of very thin coatings whether protective, lubricating, or with some other purpose (94). This, combined with new techniques such as AFM, has led to an increased interest in crystallization in thin films (95–98). Once film thickness becomes comparable with the size (eg, radius of gyration) of the polymer molecule, both the morphology and kinetics of crystallization start to change. In very thin films, where the film thickness is thinner than the thinnest stable crystal thick- ness, it is clear that material transport dominates the morphology, leading to morphological instabilities [such as the Mullins–Sekerker instability (99)], and dendritic structures (100). An example of such a structure in poly(ethylene ox- ide), in which the morphology is dominated by material transport, is shown in Figure 12. Figure 13 shows the variation in growth rate, with film thickness at a constant temperature in isotactic polystyrene (98), in which a change in the
  17. 17. Vol. 9 CRYSTALLIZATION KINETICS 481 Fig. 12. An AFM image showing part of a growing poly(ethylene oxide) dendrite in an ultrathin film on a glass substrate. A topography image, black to white represents 20 nm, is presented in pseudo 3-D. The scale bar represents 1 µm. (Unpublished data of the author). growth rate dependence once the film is thinner than the crystal thickness can be seen. Polymers can provide model systems for the study of diffusion-limited growth structures because of the tunability of the crystallization conditions, and the relatively slow rates of growth compared to small molecules (101,102). It is worth noting that under industrial processing conditions, the behavior outlined above cannot be simply applied. Typical industrial grade polymers have wide molecular weight distributions, poorly characterized degrees of branching, and possibly copolymer content. Crystallization is often carried out during a rapid quench, and so the temperature varies during crystallization and across the sam- ple. Also, crystallization often occurs at high supercoolings where, in polyethy- lene for instance, accurate growth rate data have not been obtained. The general assumption has been that the behavior during growth under these conditions is a simple superposition of the effects that have been studied under more controlled environments. However, this may not be the case, and predictive phenomenological models are only now starting to be produced for these complex but commercially important situations.
  18. 18. 482 CRYSTALLIZATION KINETICS Vol. 9 Fig. 13. A graph showing the variation of growth rate with the inverse of film thickness (1/d) for isotactic polystyrene crystals grown at 180◦ C in ultrathin films. A step in the behavior can be seen at a thickness of ∼8 nm. At this temperature, this corresponds to the thickness of the single lamella, showing a change in thickness dependence once the film is thinner than the growing crystal. Reprinted from Ref. 98. Copyright (2002), with permission from Marcel Dekker, Inc. Growth Theories Polymer crystallization theory is a mature area, and there are several review ar- ticles available that present and discuss the different theories in great detail (eg, 103,104). Having said that, over the last 5 years or so there has been a flurry of new interest because of the increase in computational power, which has the potential to decisively enter the debate in some areas. In the following the under- lying themes of the two principle theories of polymer crystallization, secondary nucleation theory and rough-surface or entropic barrier theory, are outlined. The results of more recent simulations are then briefly discussed, in which the con- straints of the above theories, introduced to provide analytical solutions, have been relaxed. Finally, some of the more fundamentally different ideas that have recently appeared are discussed. Out of the above overview of experimental data, crystallization theories have aimed primarily to explain three observations: the dependence of crystal thickness on supercooling, the dependence of crystal growth rate on supercooling, and the shape of single crystals grown from dilute solution. In all systems, crystallization occurs because of the lower free energy, over a certain temperature range, of the crystalline phase compared to the disordered
  19. 19. Vol. 9 CRYSTALLIZATION KINETICS 483 melt, solution, or vapor phase. The processes that occur as the material transforms from one state to another are very complicated and diverse. However, they can be categorized into interface and diffusion-controlled growth. In diffusion-controlled growth it is the rate of transport of something—heat, mass, etc—either to or from the growth front that limits the rate of growth. In interface-controlled growth it is the actual process of attaching and detaching molecules at the surface that controls the growth rate. In some situations both processes can have a strong in- fluence on the observed crystal morphology (100) and so it is not always simple to determine which is the rate-determining factor. In the case of polymer crystal- lization most activity has concentrated on the development of theories in which it is assumed that interface-controlled growth dominates. This is most likely a reasonable assumption in dilute solution and possibly also at small supercoolings during melt growth where faceted crystals are sometimes obtained. In both these situations it is intuitive that the physical process of attaching a long polymer molecule in register with a lattice is difficult when compared to small molecule growth. However, it is certainly not clear that it should always be the case as is often assumed. A more detailed discussion of this issue is contained in Reference 103, but suffice it to say that such theories can only be applied with confidence to solution crystallization. Secondary Nucleation Theory. Polymer secondary nucleation theory is an extension of nucleation theory in small molecules, but allowing for the connec- tivity of the units that constitute the crystal (ie, the polymer chain). In all crystal growth theories there is a driving force for crystal growth that comes from the lower free energy of the crystalline phase when compared to the liquid phase (be that melt or solution). In nucleation theory the barrier to growth is provided by an increase in surface energy that is postulated to occur when a crystallizing unit attaches in crystallographic register onto the growth front. This energy increase is due to the formation of a new crystal–liquid interface. The attachment of a single unit does not release sufficient energy to make up for the increase in free energy associated with the formation of a new surface, and so several units must form together, through random fluctuations, to provide a “nucleus.” This nucleus is a crystal patch of sufficient size that the addition of further units leads to a reduc- tion in free energy rather than to an increase. The most probable process is then for the patch to continue to grow, rather than to dissolve. The process is termed secondary nucleation because it considers the nucleation of a new patch onto an existing crystal, in contrast to primary nucleation that deals with the formation of the first nucleation event. The various nucleation theories differ in the exact way in which the attach- ment process is envisaged to occur, and in what nucleation process is presumed to be the most important (105–108). Figure 14 shows the situation that is usually considered and the definition of the different surfaces. In most nucleation theo- ries, for instance Lauritzen–Hoffmann theory (105), the nucleation process is the attachment of the first stem. This occurs in a single step [more generalized cases where it occurs in multiple steps have been considered (109–111)], and the addi- tion of further stems leads to a reduction in the total free energy of the crystal. Some justification and discussion of reducing the complex process of depositing an entire stem to a single step is given by Frank (4). Once the first stem has deposited, further increases in free energy only occur when a new fold is formed, and this has
  20. 20. 484 CRYSTALLIZATION KINETICS Vol. 9 Fig. 14. A schematic diagram representing the growth front of a polymer lamella. The polymer chain is represented by a series of boxes (stems) with defined surface energies. a high energy associated with it that can be approximated by the loss in energy that occurs from the exclusion of the fold monomers from the crystal lattice. Most forms of secondary nucleation theory assume that the chain folds back on itself into the adjacent lattice site, and so the deposition of each new stem leads to an additional fold surface energy. The rate at which stable patches nucleate on an existing crystal substrate, i, and the rate at which stable patches grow, g, controls the overall rate of crystal growth. At a particular temperature, crystals with a thickness below a certain crit- ical value (usually denoted lmin ) are unstable as, however many stems are added, the surface energy is always greater than the bulk free energy of the crystal (this is simply the Gibbs–Thomson melting point depression due to limited phase size). This minimum stable crystal thickness decreases with decreasing temperature, and gives a lower limit to the possible thickness that can grow at a particular temperature. Conversely, the rate of nucleation of stable patches, i, decreases as the thickness of the patch increases because of the increasing free-energy cost of creating the two new lateral surfaces on either side of the first stem. That is, the longer the stem the more side surface it will have and hence a higher barrier to growth. The competition between these two factors leads to a maximum in growth rate at a thickness that is slightly greater than lmin . From the above model of growth three different regimes can be defined, depending on the relative values of i and g. (1) The rate of nucleation i is considerably smaller than the rate of spreading. Each new patch grows to fill the entire available substrate before a second patch is nucleated. In this case the overall growth rate will be proportional to i. (2) The rate of nucleation i is sufficiently high that more than one growing patch exists on each substrate. Each patch is also able to support new nucleation events, and so the growth occurs in several layers. In this case the growth √ rate will be proportional to (ig) (3) A third situation can also be identified in which the rate of stem nucleation is greater than the rate of spreading, in which case the rate of growth is again proportional to i. Here the growth front has been kinetically roughened.
  21. 21. Vol. 9 CRYSTALLIZATION KINETICS 485 These three different situations have been termed regime I, II, and III re- spectively, and a large body of supporting experimental evidence on growth rates has been obtained, largely from melt crystallization, as detailed under Key Ex- perimental Results. To obtain actual values for the crystal thickness and growth rates, expres- sions for i and g need to be obtained. Here a brief outline is given, following the approach due to Lauritzen and Hoffmann (105). The change in free energy G for the formation of a patch with υ stems is given by G = − abl F + 2blσs + (υ − 1)( − abl F + 2abσe ) (7) Where σ s is the side surface free energy, σ e is the end surface free energy, l is the thickness of the crystal, a and b are the width and thickness of the stem (as marked in Fig. 14), and F is the bulk free energy of crystallization per unit volume. The term 2blσ s is due to the formation of the side surfaces, (υ − 1)(2abσ e ) is due to the end surfaces, and the terms in F are due to the reduction in free energy because of the lower energy of the bulk crystal compared to the bulk melt. In all cases it is assumed that the “known” macroscopic energies can be simply used. Four rate constants are defined: (1) A0 : the rate constant for adding a new stem to the substrate (2) B0 : the rate constant for removing an isolated stem from the substrate (3) A1 : the rate constant for adding a stem next to an existing stem of the same length (ie, without the formation of any new side surface) (4) B1 : the rate constant for removing a stem such as that added in A1 . The excess energy that comes from adding the fold is not included in the initial stem deposition (ie, process A0 ), but rather with the deposition of the ad- jacent stem. A choice is then made about how to apportion the different changes in free energy in view of the physical process of crystallization. This requires the introduction of an additional factor which defines the proportion of the bulk free energy that is released during the initial deposition of the stem and hence reduces the barrier to deposition that comes from the increase in surface free energies. From here, the steady-state flux of stems over the barrier can be determined—that is, the net rate of transition of a patch with n stems to a patch with n + 1 stems assuming that the total number of patches with n stems in the ensemble remains constant. This flux will depend on the thickness of the crystal. At a particular temperature it is assumed that there is a range of different crystal thicknesses that will grow, and the probability of obtaining a particular thickness is proportional to the steady-state flux for that crystal thickness. This leads to the average thickness of an ensemble of crystals given by 2σe l av = + δl (8) F
  22. 22. 486 CRYSTALLIZATION KINETICS Vol. 9 where δl is a small additional thickness beyond the minimum stable thickness, an exact form for which can be straightforwardly obtained (105). Thus the observed crystal thickness is slightly greater than the minimum that would be stable at the crystallization temperature, and follows the experimentally observed dependence on 1/ T. Expressions for the growth rates in the different regimes of growth can be determined by finding the values of i and g (103,105). For example, in regime I, the growth rate is given by b 2abσe −4bσs σe GI = βLp exp exp (9) a kT FkT where Lp is the persistence length of the crystal (the effective size of the crystal face onto which nucleation is occurring, usually equated with the distance between defects, rather than the total length of the crystal surface). These expressions exhibit the experimentally observed dependence of ln G on 1/T T, although an additional front factor β, which governs the rate of transport of polymer to the crystal surface, is also included. When fitting experimental data to the theory, great care needs to be taken when deciding on the form to be used for β and on the range of other behaviors that could be obtained if a different form was used. In several cases it is possible to introduce or remove multiple kinks in the growth rate curve by the choice of β, and hence to introduce types of behavior, such as growth regime changes, when none may truly exist. Secondary nucleation theory has had a great deal of success in fitting ex- perimental data, and by so doing obtaining best values for the parameters of the model. It is a highly flexible model and has been adapted to agree with most of the new experimental observations as they have been made. It does, however, suffer from several internal flaws of varying importance (see, for example, References 102 and 111 for a discussion of these) as well as, arguably, an unacceptable level of disagreement with an increasing body of experimental evidence. Whether this can be rectified within the context of a model in which the barrier to growth is nucleation, is currently unclear. Entropic Barrier Model. Although secondary nucleation theories have been very successful in fitting experimental data on growth rates and crystal thicknesses, there are several areas where, as initially formulated, they have failed. Solution-grown single crystals of polyethylene formed at low temperatures (crystallized from solvents in which there is a high entropy of mixing) have flat, crystallographic surfaces. However, if crystals are grown at higher temperatures, either from solution or in the melt, they have curved edges (113–115). If these surfaces are rough on a molecular scale, then there will be no obvious barrier to nucleation—there will always be a niche available into which a stem can deposit without having to create new side surfaces, so the initial stage of the nucleation of the first stem will never occur. It was also observed that when single crys- tals formed certain sorts of twins, a niche or reentrant corner exists at the twin boundary where a newly deposited stem would create no new side surfaces and the nucleation barrier would be removed (116). Thus, if the secondary nucleation model is correct, such twin crystals would be expected to grow very rapidly along the axis of the twin boundary. Such enhanced growth at twin boundaries is not generally observed in high polymers. These observations led Sadler and Gilmer
  23. 23. Vol. 9 CRYSTALLIZATION KINETICS 487 to reject the idea that polymer crystal growth was governed by a nucleation step, and instead to adopt a new approach, dependent on the assumption that poly- mer crystallization often occurred at sufficiently high temperatures to be above a roughening transition (117). In simple terms, the roughening transition is the temperature above which the energy cost of forming a step on the surface is less than the entropy gained by its formation. The surface then becomes rough on a macroscopic and often molecular scale. The formation of curved crystal surfaces at high temperatures is immediately explained. However, above the roughening transition, the growth rate is expected to be proportional to T, rather than the experimentally observed form ln G α 1/T T. A new barrier to growth needed to be found, and the then recently developed tool of computer simulation was utilized to do this (118–121). The thinking behind the computer simulations is as follows. Polymer units are continuously attaching and detaching from the crystal face. Growth occurs through the net accumulation of material onto the crystal face, but the possible positions in which a unit can attach are limited because of the connectivity of the polymer chain. If a stem has folded over, the stem cannot lengthen by the attachment of further units above the fold surface. If the stem is shorter than the minimum thickness it must first unfold (that is, undo the “incorrect” unit depo- sitions) before it can lengthen so as to become stable. Similarly, if a chain forms stems at two sites that are well separated, it might not be possible for them to lengthen without the detachment of one of the stems, due to the length of the connection between the two stems. Through the process of random attachment and detachment a stable crystal will grow, but its growth will be hindered by the necessity to undo, or reject, unviable unit attachments. These fluctuations that occur in and out of unviable states therefore create a barrier to growth that is entropic in origin because it arises from the consideration of all of the pathways available to the system. As the thickness of the crystal increases, the number of possible unfavorable states will increase, giving a barrier to growth that increases with thickness. Models and approximate solutions to the problem show that the barrier increases exponentially with thickness, as required by experimental obser- vation. Through Monte Carlo simulations growth rates and crystal profiles could be obtained, and these could, under certain conditions, agree with experimental observation. Analytical solutions of very simplified models could also be obtained. The principle objection to the above theory lies in its assumption of a rough- ening transition, for which little evidence exists. However, as will be seen below, the insight that a nucleation barrier is not required to explain the experimental data is an important one. New Approaches to Crystallization Kinetics Over the past decade or so there have been several significant advances both in experimental techniques, and in the power of computer simulation, that have led to an ongoing reappraisal of our understanding of polymer crystallization. In the following, some attempt is made to outline the most significant features of these new ideas and to point the interested reader toward the literature.
  24. 24. 488 CRYSTALLIZATION KINETICS Vol. 9 Computer simulation has been used as a tool to study the predictions of ex- isting theories, as well as to try to gain an insight into the controlling physics of crystal growth through direct or simplified simulation of the phase transition but with less constraint on how the process occurs (122–127). In a series of pa- pers Doye and Frenkel (128–130) have simulated both secondary nucleation and Sadler’s models but with the constraints removed that had previously been im- posed either to give analytic tractability, or sufficiently short computation time. In the simulation of the secondary nucleation model the chain is allowed to deposit unit by unit, rather than in a single step, with both attachments and detach- ments being allowed. This work shows that the proposed nucleation step, that is the deposition of the first stem, does not exist if the first two stems are deposited together—as had been previously suggested by Point (131). The peak in free en- ergy that has to be overcome for nucleation to occur is no longer seen. It is argued that this observation alone invalidates secondary nucleation theory. In simulations that are not attached to any particular theory the same au- thors have suggested a new mechanism for the selection of crystal thickness. By carrying out simulations of the deposition of new crystal layers but without any constraint that the new layer must be the same thickness as the previous layer, or indeed that each stem must have the same length as its neighbors, it was found that the growing crystal quickly adopted an average stem length that was slightly thicker than the minimum stable thickness. The mechanism behind this was quite simple. As the chain deposits, there is always a finite probability of it turning back on itself and starting to form a new, adjacent stem. If it does this before the cur- rently depositing stem has reached a stable length, it will most probably detach, and so stem lengths smaller than lmin are disallowed. However, once it is longer than lmin , the probability of growing thicker becomes decreasingly small, as there is always the same fixed probability at each new segment attachment that it will fold back to make a new stem. Crucially, the probability of a new stem being longer than the stem in the previous layer is very small as it would no longer be part of a well-defined lattice. Thus there is a particular crystal thickness at every temperature where the most probable thickness of a new layer is the same as the thickness of the previous layer, and this is the crystal thickness that will grow. In other words, there is a fixed-point attractor at a crystal thickness that is slightly thicker than the minimum stable thickness at that temperature—that is, in agreement with the experimental data. Figure 15 shows an example of a growth front formed during a simulation, and a graph showing the most probable stem length of a new stem given the length of the neighboring stem. Perhaps most interestingly, however, is the conclusion that the thickness that grows is not in fact the fastest growing at that temperature but rather the most probable. The same group reanalyzed the Sadler–Gilmer model and concluded that it was predicting a similar type of behavior. The mechanism of thickness selection in that model was also the presence of a fixed-point attractor. Simulations by Muthukumar and co-workers (133–135) have also thrown considerable doubt on some of the underlying premises of secondary nucleation theory, and also on some of the constraints that are in place in the Sadler–Gilmer model. Specifically, the assumption is generally made that each layer as formed has a particular thickness that is the final thickness of the crystal, the reorgani- zation of the chain within the crystal is not allowed for. The simulations find that
  25. 25. Vol. 9 CRYSTALLIZATION KINETICS 489 Fig. 15. (a) A cut through a polymer crystal grown in the simulation of Doye and co- workers. The crystal was produced by the growth of 20 successive layers on a surface with a uniform thickness of 50 units. The stems are represented by vertical cuboids. The cut is 16 stems wide. (b) A graph showing the dependence of the stem length on the length of the previous stem for growth of a single layer on a surface 50 units thick. There is a thickness, in this case 36 units, for which the average length of the next stem is the same as for the previous one, and this corresponds to the average thickness of the layer. This shows the way in which a new layer will converge to a particular thickness dependent on the thickness of the previous layer. Reprinted from Ref. 132. Copyright (1999), with permission from American Institute of Physics. the process of reorganization is a key part of the crystallization process, and of the selection of the final crystal thickness. The barrier to crystal growth is again found to be entropic in origin. All of these simulations apply to crystallization from solution but are still con- strained by computing power to only a relatively limited extent of crystal growth. Some of the supporting evidence in favor of secondary nucleation theory, such as the crystallographic shape, has not been dealt with explicitly. However, there clearly is considerable conflict between the existing theories and these new data, and the acceptance of a model for polymer crystal growth from solution or under similar conditions (136), in which the barrier to growth is entropic in origin, is perhaps emerging.
  26. 26. 490 CRYSTALLIZATION KINETICS Vol. 9 Fig. 15. (continued) In addition to new insights gained from computer simulation, new experi- mental techniques have led to some reassessments of previously accepted wisdom. The suggestion that spinodal decomposition of the melt could exist as a precursor to polymer nucleation has already been mentioned. Several groups have suggested that crystallization occurs through a series of intermediate stages and that the final crystal structure is not a reflection of the structure that initially formed from the supercooled melt (137–141). The ideas of the Bristol group (137,138) were based on the observation that polyethylene has a stable hexagonal phase at high temperature and pres- sure, with a hexagonal–orthorhombic–liquid triple point at a temperature of ∼215◦ C and a pressure of ∼3.6 kbar. If the polymer was crystallized at slightly lower temperatures and pressures than the triple point, the initial crystal form was still clearly hexagonal, although at a later time a transformation into the more stable orthorhombic form sometimes occurred. This led to the suggestion that, even under ambient conditions, when polymer chains attach to the growth front they initially form a crystal with hexagonal symmetry, which rapidly thick- ens in the chain direction and transforms into the orthorhombic phase when a critical thickness is reached. It was suggested that the hexagonal phase is in fact more stable than the orthorhombic phase if the crystal thickness is sufficiently small, due to the difference in the ratio of surface energy to bulk free energy. This introduced the idea of a size-dependent phase transition. On a broader basis, Ostwald’s Stage Rule (142) has been used to justify the suggestion that crystalliza- tion will occur through intermediate states if such states exist. As the material transforms from the high free-energy liquid state to the minimum free-energy
  27. 27. Vol. 9 CRYSTALLIZATION KINETICS 491 crystalline state (never reached in the case of polymers because of the kinetic trap of chain folding), it will pass through any intermediate states where local minima in free energy exist. In the case of polyethylene, the hexagonal phase in which the still mobile chains pack like parallel rods, is clearly a possible intermediate state. Direct experimental evidence of an intermediate hexagonal phase during the crystallization of polyethylene under ambient pressure has not been forth- coming. Some evidence from simulations of small molecule systems supporting the existence (and persistence) of a higher symmetry intermediate phase during nucleation has been published (143), but the complexity of polymeric systems puts similar studies beyond current computer power. A very similar set of arguments to the above has been made by Strobl and co-workers (140,141) although coming from a somewhat different set of initial assumptions. In this case it is proposed that crystallization occurs through a par- tially ordered mesophase, which then transforms into crystalline blocks and these blocks in turn heal together to form the final lamellae. Experimentally this is jus- tified by a series of observations both direct and indirect, although the primary data are a body of measurements on melting behavior and crystal thicknesses. In particular, the existence of a continuous distribution of melting temperatures for crystals of the same thickness is clear evidence of a variation in crystal perfection that is not accounted for in the traditional theories. Although these new ideas are internally consistent and supported by some experimental evidence, it has been suggested that they are contradicted strongly by the behavior of several well-studied systems. In particular, the selection of chain chirality during crystallization has been used as an argument against the possibil- ity of a loosely packed intermediate structure, both in achiral molecules where the chain can adopt helical conformations of either handedness, and in enantiomor- phic mixtures of optically pure materials. For instance, in α-iPP, where left- and right-handed helices pack in a well-ordered antiparallel fashion dictated by the stable unit-cell structure, it is hard to envisage the existence of an essentially randomly packed intermediate state that could then sort itself out into the cor- rect packing through a solid-state crystal–crystal transformation, considering the interconnectivity of the stems that make up the crystal. This argument is made very strongly in Reference 144 and is yet to be convincingly answered. Clearly there still exists considerable controversy over the factors control- ling polymer crystallization. New experimental techniques are, however, starting to become available that might lead to a resolution of some of these issues. Scan- ning probe microscopy offers the opportunity to follow crystal growth at a surface in real-time, with nanometer resolution, and hence to quantify the kinetics by direct observation at length scales that are close to those of the fundamental crys- tallizing unit, and to the size scale of the polymer chain (92,145–152). Several groups (145–147,149) have reported that lamella crystallization from the melt, at least where several lamellae are growing adjacently, does not occur at a con- stant temperature-controlled rate. Instead, growth rates are sporadic over length scales of several tens of nanometers, with lamellae or parts of lamellae spurting forwards briefly, then slowing down and being overtaken by their neighbors (146). Although this does not contradict the mean rates previously measured optically, because of the different length scale of the measurement, it does contradict the predictions of secondary nucleation theory, if it is applied to melt crystallization
  28. 28. 492 CRYSTALLIZATION KINETICS Vol. 9 (which is commonly the case; see above). Figure 16 shows a series of images of lamella crystallization, from which the growth rates can be measured. Currently it is only possible to use this technique to study crystallization at very slow rates, at the lower limits of those traditionally measured optically. However, with the ad- vent of faster scanning techniques, it should be possible to obtain real-time growth rate measurements with nanometer resolution under more usual crystallization conditions, and provide a definitive test of theory. A complimentary technique is the development of very fine beam micro-focus X-ray sources, in which submicron beams are starting to become available (153). This technique has not yet been applied to polymer crystallization, but is capable of providing spatially resolved scattering data from growth fronts that is of great Fig. 16. A series of AFM amplitude images showing the growth of an oriented polyethy- lene structure. The individual lamellae can be resolved as they grow into the melt. The growth rates of lamellae can be measured and it is seen that they vary both from lamella to lamella, and with time for each lamella. The local environment of a growing lamella is also seen to have a profound effect on its growth. (a) taken at 0 s, (b) at 181 s, (c) at 344 s, (d) at 591 s. The scale bar represents 1 µm. Reprinted from Ref. 146. Copyright (2001), with permission from the American Chemical Society.
  29. 29. Vol. 9 CRYSTALLIZATION KINETICS 493 potential value. Similarly, advances in instrumentation in other techniques such as NMR, Raman microscopy, and electron microscopy toward higher spatial and temporal resolution, as well as new methods for materials synthesis that can produce cleaner, better-defined polymers, promise to have a significant impact on our understanding of polymer crystallization kinetics at the molecular level. Conclusions The complexity of polymeric systems has made the development of a full under- standing of the crystallization process difficult. A wealth of experimental evidence has been gathered, but interpretation has often been problematic because of the ambiguities frequently found in such complex systems. The capabilities of new ex- perimental techniques to obtain both time and spatially resolved information dur- ing the phase transformation is improving rapidly and may prove to be decisive. Although secondary nucleation theory was, for a period, widely accepted, it is now coming under increasing pressure, from experimental data, from computer simulation, and from new approaches to the fundamental process of crystalliza- tion. It is not clear at this stage whether all that is required is a few adjustments to the theory, or whether the idea of a nucleation barrier is flawed, or even if the idea that the crystal thickness seen is the fastest growing is correct. With the development of new theoretical tools, and the increased integration of theory with computer simulation, it is hoped that a more complete model for polymer crystallization can be developed. A full understanding of crystallization from the melt still seems elusive, with very little work being carried out to confront the issues of mass transport, of tran- sient stresses caused by the volume reduction that accompanies crystallization [known to be sufficiently high to cause cavitation (154,155), or even to fracture the growing crystal (156)], or of the relationship between modes of growth and morphologies seen in polymers but also seen in other systems. The extent and im- portance of the role played by intermediate phases is also unclear, although this is an area that, with the development of new experimental techniques, and the push- ing of the spatial resolution of existing techniques down toward the nanometer level, should be resolvable. Despite these questions, the body of knowledge on crystallization kinetics is very extensive. As new theories are developed, it is clearly of paramount impor- tance that they use as an input all of the available information. The wide variety of different chemistries and behaviors that are observed can lead to a temptation to discard generalizing principles, but it is the author’s opinion that this temptation should be avoided. At a phenomenological level, polymer crystallization is well characterized, and the kinetics of growth can be controlled in a precise manner, at least on a macroscopic scale. It is hoped that, over the next few years, newly available experimental data can be married to the vast store of existing knowledge and assimilated, through the help of the growing power of computer simulation, to provide a new model for crystal growth that will provide the fundamental un- derstanding necessary if crystallizable polymers are to meet the demands placed on them by the emerging fields of nanotechnology and nanoscience.
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