Crystallization Kinetics

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. Becauseof 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 processingconditions, the mechanical properties, for example, Young’s modulus, of a typicalpolymer such as polyethylene can be altered by several orders of magnitude.

Because polymers crystallize so far from equilibrium conditions, a simpleexamination 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 ofthe kinetics of crystallization, as it is the kinetics of the process that define thestructure 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 routetaken by much of the research into polymer crystallization over the subsequentdecades. It soon became clear that the thickness of polymer lamellae was controlledby the supercooling at which they were crystallized and defined by the kinetics ofcrystallization. The crystal thickness, or alternatively the thickness of each newcrystalline layer in a growing crystal, is the one that grows the fastest (4,5) ratherthan the one that is at equilibrium (6). There is now a wealth of informationavailable on the crystallization of many polymers, as well as several theories thataim to predict the crystallization rates, crystal shapes, and lamellar thickness.

Crystallization kinetics is the area of polymer science that deals with therate at which randomly ordered chains transform into highly ordered crystals, andincludes every aspect of the resultant structure that is dependent on the route thatwas taken between those different states. It is a broad and mature area of scien-tific research, given an uncommon diversity when compared to the crystallizationof small molecules because of the wide range of different chemistries and chaintopologies that are available to macromolecules. These add layers of complexitythat 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 theauthor, and to principles and observations that have, in the author’s opinion, thewidest applicability.

As in any transformation, it is possible to measure bulk rates of polymercrystallization and to characterize the process in terms of the shape of thesetransformation curves. Because of limitations on space the interested reader ispointed toward the relevant literature (7–12), as this approach, although usefulfor quantifying experimental observations, does not attempt to provide an insightinto what is happening at the molecular scale. Polymer crystallization, like manyphase transitions, occurs through a process of nucleation and growth, and it isoften 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 offundamental importance, the nucleation step is still poorly understood and only

Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

466 CRYSTALLIZATION KINETICS Vol. 9

occasionally studied. The next section outlines the key experimental data and thetheories that have been developed to explain this data. A brief look at some morerecent approaches is also given. The majority of this article considers the growthprocess itself, concentrating on linear, flexible polymers as this is where the mostcomplete data set is available, but also touching on more complex materials. Anoutline of the theories that have been developed to explain and predict crystal-lization behavior is also given, concentrating on their fundamental assumptionsand predictions rather than their mathematical intricacies.

Nucleation

Any supercooled or supersaturated system contains local fluctuations in orderand 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 isthe size at which it is in unstable equilibrium with the surrounding phase—it hasformed a critical nucleus, and there is a high probability of it continuing to growto macroscopic size.

Nucleation can be divided into homogeneous nucleation and heterogeneousnucleation, the former being the sporadic formation of critical nuclei from the purephase, the latter occurring at the surface of impurities within the system. In highpolymers heterogeneous nucleation dominates in most cases (13). Heterogeneousnucleation can be divided into different classes; epitaxy, in which there is a latticematch [usually within 10% (14,15)] between a crystal plane in the polymer anda free surface in the “impurity”, and nonspecific surface nucleation, in which thepresence of the impurity surface leads to a chain conformation closer to that in thecrystal. A final class of nucleation is self-nucleation, in which during the melting ofthe 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 iscaused by the broad range of melting temperatures typically present in any samplebecause of a combination of factors including the range of crystal sizes, molecularweight, chain perfection, and local environment. Small crystalline “seeds” are leftin 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 canbe seen that the individual spherulites intersect with each other to form polygonswith straight sides. It is immediately apparent that, assuming the growth rate isconstant across the sample, all of the nucleation sites must have become active atthe 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 underthese most common circumstances, occurring on impurities or added nucleating


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 crystallizationis 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 numberof nuclei per unit volume, and the variation of this with temperature can be sim-ply obtained (20). Some polymers behave differently, nucleation occurring moresporadically, giving a measurable nucleation rate as a function of crystallizablevolume at a particular temperature (21,22). An example of this behavior is shownin Figure 2 for polyethylene succinate, with the variation with molecular weightalso included. Whether this nucleation is actually homogeneous, or if the numberof 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 figurefor the side surface free energy for use in secondary nucleation crystal growththeory. 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


Fig. 2. A graph showing the temperature dependence of the nucleation rate for a seriesof different molecular weight fractions of polyethylene succinate. The molecular weightshown 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 relativelyearly 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 nucleationrate with temperature can be obtained. If the nucleation is truly homogeneous,and the droplet size is monodisperse, the nucleation rate should, at a particulartemperature, depend only on the sample volume. Recently, similar experimentshave 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 inFigures 3 and 4.

Nucleation Theory. Classical theories for nucleation in small moleculesbalance the reduction in free energy that occurs because the solid is more stablethan 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 isgiven by

G = 43

πr3 Gυ + 4πr2σ (1)

Where G is the Gibbs free energy, r is the radius of the sphere, Gυ isthe 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 Ggoes through a maximum at the critical nucleus size, resulting in a nucleationbarrier—the maximum in the increase in free energy necessary to form the critical


Fig. 3. A composite graph showing the crystallization of micro phase-separated spheres ofE/SEB63 (polyethylene block styrene–ethylene–butene). The polyethylene block exists inspherical domains with a radius of 12.7 nm in a matrix of rubbery material. At sufficientlyhigh supercoolings the polyethylene nucleates separately in each domain. The graph showsthe 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 (theinsets 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 fromthe American Physical Society.

nucleus—given by

G∗ = 16πσ 3

3( Gυ)2(2)

For polymers the situation is changed by two factors. Firstly, the connectivityalong the chain of identical crystallizing units means that the nucleus is unlikely tobe 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 theside surface. These lead to a slightly different equation, allowing for the differentgeometry, given by

G = −υµla0b0 Gυ + 2µυa0b0σe + 2σ l(µb0 + υa0) (3)


Fig. 4. A graph showing the rate of nucleation of droplets of polyethylene. Isolateddroplets 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 supercoolingscan be reached. The nucleation rate can be measured directly by counting the numberof droplets that have crystallized as a function of time. The data were all collected fromthe same preparation of droplets, which could not be cooled below 87◦C without all thedroplets 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 thelamella, a0 and b0 are the cross-sectional dimensions of a stem, l is the thicknessof the lamella, Gν is the difference in bulk free energy between the crystallineand liquid phases, σ e is the fold surface free energy, and σ is the side surface freeenergy. This then leads to the rate of formation I of folded chain nuclei given by

I = I0 exp[ − FD

kT

]exp

[32σ 2σe

(T0

m

)2kT( T)2( Hυ)2

](4)

which is equivalent to the well-known Turnbull–Fisher equation (30). The fac-tor FD here corresponds to the energy barrier affecting transport of materialacross 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, Hν is the enthalpy of fusionper unit volume, T0

m is the equilibrium melting temperature, and k is the Boltz-mann constant. Heterogeneous nucleation is introduced simply by altering one ofthe side surface energies to allow for the relatively low energy penalty associatedwith the formation of a polymer-nucleating surface interface (13). Experimentalstudies (19,24–27) are broadly in agreement with the predicted behavior for the


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 highmolecular 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 asthe concentration is increased, or in the melt. An alternative is to assume thatthe nucleation is of a bundle of stems, each stem coming from a different polymerchain, forming a ciliated or “fringed-micelle” nucleus. Now, in addition to the un-favorable entropy contribution from demixing the polymer from solution, there isan unfavorable contribution from the restrictions on the conformations accessibleto the noncrystalline portion of the polymer molecules imposed by their incorpo-ration into the nucleus. Flory (31) [and later Mandelkern (32)] considered thesecontributions in his original work on polymer crystallization, and, although someof the initial assumptions have proven to be in disagreement with experiment, theresulting formalism is still relevant under certain situations such as the crystal-lization of parts of chains confined between lamellae. This approach leads to thefollowing expression for the free energy of formation of a ciliated crystal, in thelimit of high dilution (polymer volume fraction tends to zero):

G = −υµla0b0 Gυ + 2µυa0b0

(σe − kT lnν2

2a0b0

)+ 2σ l(µb0 + υa0) (5)

where ν2 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 intosuch small volumes in order to observe something that might be homogeneousnucleation that the effect of the surfaces of the droplet might well start to play asignificant role (33,34). Studies using block copolymers, although of great value,are studying nucleation in a different polymer, and the effect of connectivity to anuncrystallizable unit is complex (35).

Recently, several groups have reported data collected in the early stages ofcrystal growth that they have interpreted as being evidence of spinodal decom-position in the melt prior to nucleation and growth (36–38). These follow fromsimilar suggestions made in the 1980s (39–41). The interpretations are basedaround the observation of small-angle X-ray scattering (SAXS), usually associ-ated with the formation of lamellae, prior to the wide-angle signal that comesfrom crystallographic order. The evolution of the SAXS signal in this initial stageis characterized by first-order kinetics (Avrami exponent of one), and identifiedwith a spinodal-like transformation. This is a difficult area, as identifying thepoint 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- andsmall-angle data (38). It has been proposed that the components of the melt thatare phase-separating are segments of the chain that are closer to, and furtherfrom, their crystalline conformation. This separation is driven by a coupling be-tween density and order in the chain, in that by aligning, for instance, similarlyhelical chains they are able to pack more closely and increase the local density


(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 orderedregions. The “phase” separation drives a change in order into a state that is stillfar from crystalline, but has a higher probability of crystallizing than the initiallysupercooled melt. If the transformation occurs under conditions of constant pres-sure, as is usually the case, it will eventually result in an overall densification ofthe melt. The theory is internally consistent, but there is still an active debateas to whether or not it actually applies to the experimental situation. A priori, itseems more likely that it is of relevance to polymers with more rigid chains, andcrystallization at high supercoolings.

Crystal Growth

In this section an overview of key experimental results on the growth of polymercrystals is given, concentrating on linear molecules with simple chemistry andmoderate molecular weight. Although these molecules are the most commonlyused in industrial applications, there are many other types of behavior that aremarkedly different, and some indication of these perturbing effects is given. Thiswill provide the data against which any theory should be tested, and it is thebasis of the different theories that are discussed in the next section. Finally, overthe 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 scanningprobe microscopy and to the increasing power of computer simulation. These newapproaches 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 basalplane (1). As these lamellae are typically between 5 and 50 nm in thickness, thechains must fold back on themselves, reentering the same crystal many times.The extent to which chains reenter the crystal on average at the adjacent latticesite, or elsewhere, has been a subject of argument for 40 years (44). Although theexact 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 polymercrystallization. It has been widely accepted since the early 1960s that the equi-librium polymer crystal contains chains that are extended. The lamellae that arecommonly 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 crystalsrather 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 crystalthickness, but rather the supercooling (or supersaturation) at which crystalliza-tion occurred. Supercooling is here defined as the equilibrium melting tempera-ture (T0

m) minus the crystallization temperature, where the equilibrium meltingtemperature is the temperature at which an equilibrium, infinite, extended chaincrystal would melt. These data were collected for polyethylene, but similar dataexist for a range of other polymers. The supercooling controls the crystal thickness,


Fig. 5. A graph showing the variation of the fold length (l) with supercooling ( T) forpolyethylene crystallized from a variety of solvents and from the melt. In the case of solventcrystallization, supercooling is taken with respect to the so-called equilibrium dissolutiontemperature. For the melt-crystallized data set the equilibrium melting temperature isused. The remarkable coincidence between the curves, despite the wide range of absolutetemperatures to which each supercooling corresponds, is strong evidence in favor of thekinetic origin of crystal thickness selection. Solvents: � xylene, � hexyl acetate, ⊕ ethylesters, � dodecanol, � dodecane, � octane, × tetradecanol, + hexadecane, � melt crystal-lized. Reprinted from Ref. 44. Copyright (1985), with permission from Kluwer AcademicPublishers.

and the thickness that grows is just thicker than the minimum stable thicknessat that temperature. This minimum stable thickness comes from the balancing ofthe increase in energy due to the presence of the crystal melt interface, with thereduction in free energy due to the bulk free energy of the crystal—that is, theGibbs–Thomson equation. In the case of lamellae with large lateral dimensionsthis can be simplified to give

lmin = 2σe

F(6)

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 thenoncrystalline 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).


Fig. 6. A graph of the variation in growth rate with temperature forpoly(hydroxybutyrate), showing the typically observed bell-shaped temperature de-pendence, with the crystallization rate reducing to zero well before the equilibriummelting temperature (∼198◦C for this polymer) and, after passing through a maximum,again reducing to zero before the glass-transition temperature (∼0◦C). (Unpublished dataof the author.)

As detailed in the article on semicrystalline polymers, polymer lamellae,when crystallized from the melt, typically form aggregates. The most common ofthese aggregates, if the melt is unperturbed, is the spherulite. The growth ratesof spherulites have been measured as a function of crystallization temperaturefor a wide range of different polymers. These rates are found to be approximatelyconstant at a particular temperature for a particular polymer sample. An exam-ple of the variation in growth rate with temperature of polyhydroxybutyrate, abiodegradable thermoplastic, is shown in Figure 6. A bell-shaped relationshipis apparent, typical of glass-forming liquids. Similar data have been obtainedfor many other polymers, the lower limit of the growth depending on the glass-transition temperature of the material, the upper limit depending on the meltingtemperature, and the peak rate varying from only a few nanometers per second tohundreds of micrometers per second, depending on the polymer. The inset showsan optical micrograph of a growing spherulite from which rate measurements canbe made. The rate that is measured in a typical experiment is the average growthrate of many lamellae as they grow over a distance of several microns. It is not thegrowth rate of an individual lamella, and is certainly not measured over lengthscales comparable with either the size of the unit cell, or even the thickness of alamella. 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 nucleationtheory (see section under Secondary Nucleation Theory). A large body of data


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 thecrystal morphology. A systematic survey of the effect of varying the additionaltransport term that is often included in the y-axis of the plot, or of the often im-precisely defined value of Tm

0, is not available. However, growth rate is clearlyexponentially 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 ofpolyethylene succinate (62). The variation in growth rate with molecular weightleads 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 fora series of different molecular weight fractions of polyethylene succinate, showing the typ-ically observed behavior. Again, the molecular weight shown is the peak molecular weightas measured by GPC. In Ref. 61 the authors obtain a “master curve” for polymer growthrate by plotting the reduced growth rate G/Gmax (where G is the growth rate and Gmax is thefastest growth rate exhibited by the particular polymer sharp fraction) against the reducedtemperature T/Tcmax (where T is the crystallization temperature and Tcmax is the temper-ature at which the maximum growth rate occurs). This leads to the intriguing possibilityof a universal relationship between growth rate and molecular weight. In particular, it isfound that the molecular weight dependence of Gmax can be expressed as a power law, Gmaxproportional to MWα where α depends on the adsorption mechanism for polymer moleculeson 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 fromRef. 62. Copyright (2002), with permission from Elsevier Science Ltd.


samples there is a distribution of chains with different molecular lengths. Carefulstudies of the molecular weight distribution within the crystallized and uncrystal-lized portions of a sample at different stages of growth have shown a remarkabledegree of fractionation (63,64). At high temperatures the longest chains crystal-lize most readily and so fractionation occurs during growth, leaving the shortermolecules to either crystallize more slowly later, or on cooling if the distribution oflengths 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 havea stronger influence on the measured growth rate. The extent of fractionation,particularly apparent during solution crystallization, has lead to the suggestionthat there must be a process of “molecular nucleation” (65), in which each moleculenucleates on the growth front as a separate object rather than the important stepbeing the nucleation of parts of a chain or chains. This provides a mechanismby which the growth process can “feel” the molecular length of each chain thatattaches to the growth front. A fuller discussion is given in Reference 65.

Crystallization studies carried out in dilute solution have frequently beenused to gain additional insights into the process of growth, as the crystallizinglamellae can be more straight forwardly examined by electron microscopy aftertheir 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 regularcrystal thickness can be seen. The well-defined crystallographic shape is typicalof solution-grown crystals of many (but not all) polymers, their exact shape de-pending on the underlying crystallography and the interplay of growth rates ofthe different crystal faces, which also varies with crystallization temperature (seeSEMICRYSTALLINE POLYMERS for more detail on crystal shape). Crystal thickness istypically plotted against the equilibrium dissolution temperature, a temperaturethat has an ill-defined meaning (68,69), but these plots give a remarkably similartemperature dependence of thickness to that seen for melt crystallization, in thecase of polyethylene. Despite technical difficulties, growth rates as a function oftemperature can be measured by careful experimentation (70–75), and can beplotted to give an approximately linear relationship with 1/T T. An example isshown 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 beyondthe scope of this review.

There are many factors that can affect the way crystallization occurs, andlead to differences from the above outlined pattern of behavior. Linear homopoly-mers consist of repeating units, all of which can crystallize. Even in this simplecase, the end groups are chemically different from the rest of the chain, and thiscan lead to some perturbations in behavior. End groups are usually excluded tothe surface of crystals (76). For low molecular weight polymers (Mw < 10,000), thepresence 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 thatthe 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 growthrate are observed in the region where the crystallization temperature allowsthinner crystals to become stable (80). An example of these local minima in growth


Fig. 8. An AFM topographic image showing a group of single crystals of polyethylenecrystallized from xylene at 70◦C. The grey scale shows the variation in height, from whichthe very uniform thickness of the crystal can be clearly seen. (Unpublished data; imagecourtesy of Dr. A. K. Winkel).

rate is shown in Figure 10 (81). These minima occur because of competition at thegrowth front between different metastable “phases”—for example once-folded andtwice-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 ofactive debate. However, the recent advent of more controlled techniques for poly-merization offers hope of reaching a greater understanding. In most cases, one ofthe copolymer species will crystallize, excluding the other from its lattice. Crys-tals can only form if sequence lengths are sufficiently long to span the thicknessof 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 asto access the crystallizable sequences, and this additional step slows the growth


Fig. 9. A graph showing the variation in growth rate with 1/T T for single crystalsof polyethylene grown from different solvents. The top data set is for 0.05% Rigidex 50 intetradecanol, 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 thehigher temperature lines is exactly 2, as predicted by theory (see section under SecondaryNucleation Theory). Reprinted from Ref. 74. Copyright (1986), with permission from JohnWiley & 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 arenot stable at that temperature), and so will be unable to crystallize until a lowertemperature is reached. Lower total crystallinities are usually obtained, becauseit is not always possible for all of the sequences that are potentially crystallizableat a particular temperature to find a crystal into which to crystallize, consideringthe constraints due to the crystallization of other sequences in the same chain.A very similar behavior is observed with branched polymers, as longer branches


Fig. 10. A graph showing the variation in crystallization rate with temperature for theultralong alkane C246H494. + 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 isbelieved to be caused by competition at the growth front between the more stable thickercrystal 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 themaximum crystal thickness into which a particular chain segment can crystal-lize. If this is thinner than the minimum stable thickness for a crystal, the chainsegment cannot crystallize at that temperature. Again, this leads to a slowing ofthe overall growth rate, a reduction in the achieved crystallinity, and a wideningof the temperature over which crystallization occurs. In the case of polyethylene,which has been extensively studied because of the commercial importance of lowdensity 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 hexagonalphase.

As well as chain chemistry and chain length, different behaviors from theabove outlined “typical behavior” can also be found because of differences in thecrystallization conditions of an otherwise “typical” flexible linear polymer. Crys-tallization under flow leads to a change in morphology (87–89). Initially, orientednuclei are formed with a largely extended chain character. These nuclei consistof the high molecular weight tail of the molecular weight distribution, both highmolecular weight and polydispersity apparently being required (89–91). Fromthese oriented backbones, plate-like lamellae then grow, largely perpendicular tothe 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 geometriesfor the study of lamellar growth (93). However, the growth of the backbone itselfis very difficult to follow, due both to its size—although up to several microns inlength, their width is typically less than 10 nm—and the very rapid rate of itsformation.


Fig. 11. An AFM micrograph (phase image) showing a shish-kebab crystal of polyethylenein a matrix of molten polymer. Image collected at 135◦C. The bright area in the top right ofthe image is the glass substrate. The extended chain backbone and lamellar overgrowthscan 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 valuethrough the use of very thin coatings whether protective, lubricating, or with someother purpose (94). This, combined with new techniques such as AFM, has led toan increased interest in crystallization in thin films (95–98). Once film thicknessbecomes comparable with the size (eg, radius of gyration) of the polymer molecule,both the morphology and kinetics of crystallization start to change. In very thinfilms, where the film thickness is thinner than the thinnest stable crystal thick-ness, it is clear that material transport dominates the morphology, leading tomorphological instabilities [such as the Mullins–Sekerker instability (99)], anddendritic structures (100). An example of such a structure in poly(ethylene ox-ide), in which the morphology is dominated by material transport, is shown inFigure 12. Figure 13 shows the variation in growth rate, with film thickness ata constant temperature in isotactic polystyrene (98), in which a change in the


Fig. 12. An AFM image showing part of a growing poly(ethylene oxide) dendrite in anultrathin film on a glass substrate. A topography image, black to white represents 20 nm, ispresented 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 canbe seen. Polymers can provide model systems for the study of diffusion-limitedgrowth structures because of the tunability of the crystallization conditions, andthe relatively slow rates of growth compared to small molecules (101,102).

It is worth noting that under industrial processing conditions, the behavioroutlined above cannot be simply applied. Typical industrial grade polymers havewide molecular weight distributions, poorly characterized degrees of branching,and possibly copolymer content. Crystallization is often carried out during a rapidquench, 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 generalassumption has been that the behavior during growth under these conditionsis a simple superposition of the effects that have been studied under morecontrolled environments. However, this may not be the case, and predictivephenomenological models are only now starting to be produced for these complexbut commercially important situations.


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 thebehavior can be seen at a thickness of ∼8 nm. At this temperature, this corresponds tothe thickness of the single lamella, showing a change in thickness dependence once thefilm is thinner than the growing crystal. Reprinted from Ref. 98. Copyright (2002), withpermission 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 flurryof new interest because of the increase in computational power, which has thepotential to decisively enter the debate in some areas. In the following the under-lying themes of the two principle theories of polymer crystallization, secondarynucleation theory and rough-surface or entropic barrier theory, are outlined. Theresults of more recent simulations are then briefly discussed, in which the con-straints of the above theories, introduced to provide analytical solutions, havebeen relaxed. Finally, some of the more fundamentally different ideas that haverecently appeared are discussed.

Out of the above overview of experimental data, crystallization theories haveaimed primarily to explain three observations: the dependence of crystal thicknesson supercooling, the dependence of crystal growth rate on supercooling, and theshape of single crystals grown from dilute solution.

In all systems, crystallization occurs because of the lower free energy, overa certain temperature range, of the crystalline phase compared to the disordered


melt, solution, or vapor phase. The processes that occur as the material transformsfrom one state to another are very complicated and diverse. However, they can becategorized into interface and diffusion-controlled growth. In diffusion-controlledgrowth it is the rate of transport of something—heat, mass, etc—either to or fromthe growth front that limits the rate of growth. In interface-controlled growth itis the actual process of attaching and detaching molecules at the surface thatcontrols 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 simpleto determine which is the rate-determining factor. In the case of polymer crystal-lization most activity has concentrated on the development of theories in whichit is assumed that interface-controlled growth dominates. This is most likely areasonable assumption in dilute solution and possibly also at small supercoolingsduring melt growth where faceted crystals are sometimes obtained. In both thesesituations it is intuitive that the physical process of attaching a long polymermolecule in register with a lattice is difficult when compared to small moleculegrowth. However, it is certainly not clear that it should always be the case as isoften assumed. A more detailed discussion of this issue is contained in Reference103, but suffice it to say that such theories can only be applied with confidence tosolution crystallization.

Secondary Nucleation Theory. Polymer secondary nucleation theory isan 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 crystalgrowth theories there is a driving force for crystal growth that comes from thelower free energy of the crystalline phase when compared to the liquid phase (bethat melt or solution). In nucleation theory the barrier to growth is provided byan increase in surface energy that is postulated to occur when a crystallizing unitattaches in crystallographic register onto the growth front. This energy increase isdue to the formation of a new crystal–liquid interface. The attachment of a singleunit does not release sufficient energy to make up for the increase in free energyassociated with the formation of a new surface, and so several units must formtogether, through random fluctuations, to provide a “nucleus.” This nucleus is acrystal 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 thenfor the patch to continue to grow, rather than to dissolve. The process is termedsecondary nucleation because it considers the nucleation of a new patch onto anexisting crystal, in contrast to primary nucleation that deals with the formationof 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 presumedto be the most important (105–108). Figure 14 shows the situation that is usuallyconsidered and the definition of the different surfaces. In most nucleation theo-ries, for instance Lauritzen–Hoffmann theory (105), the nucleation process is theattachment of the first stem. This occurs in a single step [more generalized caseswhere 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 anentire 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


Fig. 14. A schematic diagram representing the growth front of a polymer lamella. Thepolymer 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 energythat occurs from the exclusion of the fold monomers from the crystal lattice. Mostforms of secondary nucleation theory assume that the chain folds back on itselfinto the adjacent lattice site, and so the deposition of each new stem leads to anadditional 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 crystalgrowth. 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 (thisis 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 particulartemperature. Conversely, the rate of nucleation of stable patches, i, decreases asthe thickness of the patch increases because of the increasing free-energy cost ofcreating the two new lateral surfaces on either side of the first stem. That is, thelonger the stem the more side surface it will have and hence a higher barrier togrowth. The competition between these two factors leads to a maximum in growthrate 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 secondpatch is nucleated. In this case the overall growth rate will be proportionalto i.

(2) The rate of nucleation i is sufficiently high that more than one growing patchexists on each substrate. Each patch is also able to support new nucleationevents, and so the growth occurs in several layers. In this case the growthrate will be proportional to

√(ig)

(3) A third situation can also be identified in which the rate of stem nucleation isgreater than the rate of spreading, in which case the rate of growth is againproportional to i. Here the growth front has been kinetically roughened.


These three different situations have been termed regime I, II, and III re-spectively, and a large body of supporting experimental evidence on growth rateshas 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 theapproach due to Lauritzen and Hoffmann (105).

The change in free energy G for the formation of a patch with υ stems isgiven 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 unitvolume. 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 freeenergy 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 simplyused. 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 theinitial 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 changesin free energy in view of the physical process of crystallization. This requires theintroduction of an additional factor � which defines the proportion of the bulk freeenergy that is released during the initial deposition of the stem and hence reducesthe barrier to deposition that comes from the increase in surface free energies.

From here, the steady-state flux of stems over the barrier can bedetermined—that is, the net rate of transition of a patch with n stems to a patchwith n + 1 stems assuming that the total number of patches with n stems in theensemble 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 crystalthicknesses that will grow, and the probability of obtaining a particular thicknessis proportional to the steady-state flux for that crystal thickness. This leads to theaverage thickness of an ensemble of crystals given by

〈l〉av = 2σe

F+ δl (8)


where δl is a small additional thickness beyond the minimum stable thickness, anexact form for which can be straightforwardly obtained (105). Thus the observedcrystal thickness is slightly greater than the minimum that would be stable at thecrystallization temperature, and follows the experimentally observed dependenceon 1/ T. Expressions for the growth rates in the different regimes of growth canbe determined by finding the values of i and g (103,105). For example, in regimeI, the growth rate is given by

GI = ba

βLp exp(

2abσe�

kT

)exp

( −4bσsσe

FkT

)(9)

where Lp is the persistence length of the crystal (the effective size of the crystalface onto which nucleation is occurring, usually equated with the distance betweendefects, rather than the total length of the crystal surface). These expressionsexhibit the experimentally observed dependence of ln G on 1/T T, although anadditional front factor β, which governs the rate of transport of polymer to thecrystal 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 onthe 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 growthrate curve by the choice of β, and hence to introduce types of behavior, such asgrowth 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 themodel. It is a highly flexible model and has been adapted to agree with most of thenew experimental observations as they have been made. It does, however, sufferfrom several internal flaws of varying importance (see, for example, References102 and 111 for a discussion of these) as well as, arguably, an unacceptable levelof disagreement with an increasing body of experimental evidence. Whether thiscan be rectified within the context of a model in which the barrier to growth isnucleation, is currently unclear.

Entropic Barrier Model. Although secondary nucleation theories havebeen very successful in fitting experimental data on growth rates and crystalthicknesses, there are several areas where, as initially formulated, they havefailed. 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 thesesurfaces are rough on a molecular scale, then there will be no obvious barrier tonucleation—there will always be a niche available into which a stem can depositwithout having to create new side surfaces, so the initial stage of the nucleationof 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 twinboundary where a newly deposited stem would create no new side surfaces andthe nucleation barrier would be removed (116). Thus, if the secondary nucleationmodel is correct, such twin crystals would be expected to grow very rapidly alongthe axis of the twin boundary. Such enhanced growth at twin boundaries is notgenerally observed in high polymers. These observations led Sadler and Gilmer


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 aroughening transition (117).

In simple terms, the roughening transition is the temperature above whichthe energy cost of forming a step on the surface is less than the entropy gainedby its formation. The surface then becomes rough on a macroscopic and oftenmolecular scale. The formation of curved crystal surfaces at high temperaturesis immediately explained. However, above the roughening transition, the growthrate is expected to be proportional to T, rather than the experimentally observedform ln G α 1/T T. A new barrier to growth needed to be found, and the thenrecently developed tool of computer simulation was utilized to do this (118–121).

The thinking behind the computer simulations is as follows. Polymer unitsare continuously attaching and detaching from the crystal face. Growth occursthrough the net accumulation of material onto the crystal face, but the possiblepositions in which a unit can attach are limited because of the connectivity ofthe polymer chain. If a stem has folded over, the stem cannot lengthen by theattachment of further units above the fold surface. If the stem is shorter than theminimum 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 formsstems at two sites that are well separated, it might not be possible for them tolengthen without the detachment of one of the stems, due to the length of theconnection between the two stems. Through the process of random attachmentand detachment a stable crystal will grow, but its growth will be hindered by thenecessity to undo, or reject, unviable unit attachments. These fluctuations thatoccur in and out of unviable states therefore create a barrier to growth that isentropic in origin because it arises from the consideration of all of the pathwaysavailable to the system. As the thickness of the crystal increases, the number ofpossible unfavorable states will increase, giving a barrier to growth that increaseswith thickness. Models and approximate solutions to the problem show that thebarrier increases exponentially with thickness, as required by experimental obser-vation. Through Monte Carlo simulations growth rates and crystal profiles couldbe obtained, and these could, under certain conditions, agree with experimentalobservation. 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 experimentaldata is an important one.

New Approaches to Crystallization Kinetics

Over the past decade or so there have been several significant advances both inexperimental techniques, and in the power of computer simulation, that have ledto an ongoing reappraisal of our understanding of polymer crystallization. In thefollowing, some attempt is made to outline the most significant features of thesenew ideas and to point the interested reader toward the literature.


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 physicsof crystal growth through direct or simplified simulation of the phase transitionbut 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 andSadler’s models but with the constraints removed that had previously been im-posed either to give analytic tractability, or sufficiently short computation time. Inthe simulation of the secondary nucleation model the chain is allowed to depositunit 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 isthe deposition of the first stem, does not exist if the first two stems are depositedtogether—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 arguedthat 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. Bycarrying out simulations of the deposition of new crystal layers but without anyconstraint that the new layer must be the same thickness as the previous layer, orindeed that each stem must have the same length as its neighbors, it was foundthat the growing crystal quickly adopted an average stem length that was slightlythicker than the minimum stable thickness. The mechanism behind this was quitesimple. As the chain deposits, there is always a finite probability of it turning backon 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 longerthan lmin, the probability of growing thicker becomes decreasingly small, as thereis always the same fixed probability at each new segment attachment that it willfold back to make a new stem. Crucially, the probability of a new stem beinglonger than the stem in the previous layer is very small as it would no longerbe part of a well-defined lattice. Thus there is a particular crystal thickness atevery temperature where the most probable thickness of a new layer is the sameas the thickness of the previous layer, and this is the crystal thickness that willgrow. In other words, there is a fixed-point attractor at a crystal thickness thatis slightly thicker than the minimum stable thickness at that temperature—thatis, in agreement with the experimental data. Figure 15 shows an example of agrowth front formed during a simulation, and a graph showing the most probablestem length of a new stem given the length of the neighboring stem. Perhaps mostinterestingly, however, is the conclusion that the thickness that grows is not in factthe fastest growing at that temperature but rather the most probable. The samegroup reanalyzed the Sadler–Gilmer model and concluded that it was predictinga similar type of behavior. The mechanism of thickness selection in that modelwas also the presence of a fixed-point attractor.

Simulations by Muthukumar and co-workers (133–135) have also thrownconsiderable doubt on some of the underlying premises of secondary nucleationtheory, and also on some of the constraints that are in place in the Sadler–Gilmermodel. Specifically, the assumption is generally made that each layer as formedhas 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


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 witha uniform thickness of 50 units. The stems are represented by vertical cuboids. The cut is16 stems wide. (b) A graph showing the dependence of the stem length on the length of theprevious 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 theprevious one, and this corresponds to the average thickness of the layer. This shows the wayin which a new layer will converge to a particular thickness dependent on the thicknessof the previous layer. Reprinted from Ref. 132. Copyright (1999), with permission fromAmerican Institute of Physics.

the process of reorganization is a key part of the crystallization process, and ofthe selection of the final crystal thickness. The barrier to crystal growth is againfound 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, suchas the crystallographic shape, has not been dealt with explicitly. However, thereclearly is considerable conflict between the existing theories and these new data,and the acceptance of a model for polymer crystal growth from solution or undersimilar conditions (136), in which the barrier to growth is entropic in origin, isperhaps emerging.


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 precursorto polymer nucleation has already been mentioned. Several groups have suggestedthat crystallization occurs through a series of intermediate stages and that thefinal crystal structure is not a reflection of the structure that initially formed fromthe supercooled melt (137–141).

The ideas of the Bristol group (137,138) were based on the observationthat 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 slightlylower temperatures and pressures than the triple point, the initial crystal formwas still clearly hexagonal, although at a later time a transformation into themore stable orthorhombic form sometimes occurred. This led to the suggestionthat, even under ambient conditions, when polymer chains attach to the growthfront they initially form a crystal with hexagonal symmetry, which rapidly thick-ens in the chain direction and transforms into the orthorhombic phase when acritical thickness is reached. It was suggested that the hexagonal phase is in factmore stable than the orthorhombic phase if the crystal thickness is sufficientlysmall, 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 materialtransforms from the high free-energy liquid state to the minimum free-energy


crystalline state (never reached in the case of polymers because of the kinetic trapof chain folding), it will pass through any intermediate states where local minimain free energy exist. In the case of polyethylene, the hexagonal phase in whichthe still mobile chains pack like parallel rods, is clearly a possible intermediatestate. Direct experimental evidence of an intermediate hexagonal phase duringthe crystallization of polyethylene under ambient pressure has not been forth-coming. Some evidence from simulations of small molecule systems supportingthe existence (and persistence) of a higher symmetry intermediate phase duringnucleation has been published (143), but the complexity of polymeric systems putssimilar studies beyond current computer power.

A very similar set of arguments to the above has been made by Strobl andco-workers (140,141) although coming from a somewhat different set of initialassumptions. In this case it is proposed that crystallization occurs through a par-tially ordered mesophase, which then transforms into crystalline blocks and theseblocks 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 primarydata are a body of measurements on melting behavior and crystal thicknesses. Inparticular, the existence of a continuous distribution of melting temperatures forcrystals of the same thickness is clear evidence of a variation in crystal perfectionthat is not accounted for in the traditional theories.

Although these new ideas are internally consistent and supported by someexperimental evidence, it has been suggested that they are contradicted stronglyby the behavior of several well-studied systems. In particular, the selection of chainchirality during crystallization has been used as an argument against the possibil-ity of a loosely packed intermediate structure, both in achiral molecules where thechain can adopt helical conformations of either handedness, and in enantiomor-phic mixtures of optically pure materials. For instance, in α-iPP, where left- andright-handed helices pack in a well-ordered antiparallel fashion dictated by thestable unit-cell structure, it is hard to envisage the existence of an essentiallyrandomly packed intermediate state that could then sort itself out into the cor-rect packing through a solid-state crystal–crystal transformation, considering theinterconnectivity of the stems that make up the crystal. This argument is madevery 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, startingto 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 surfacein real-time, with nanometer resolution, and hence to quantify the kinetics bydirect 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). Severalgroups (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 lengthscales of several tens of nanometers, with lamellae or parts of lamellae spurtingforwards 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 thepredictions of secondary nucleation theory, if it is applied to melt crystallization


(which is commonly the case; see above). Figure 16 shows a series of images oflamella crystallization, from which the growth rates can be measured. Currentlyit 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 growthrate measurements with nanometer resolution under more usual crystallizationconditions, and provide a definitive test of theory.

A complimentary technique is the development of very fine beam micro-focusX-ray sources, in which submicron beams are starting to become available (153).This technique has not yet been applied to polymer crystallization, but is capableof 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. Thegrowth rates of lamellae can be measured and it is seen that they vary both from lamellato lamella, and with time for each lamella. The local environment of a growing lamella isalso seen to have a profound effect on its growth. (a) taken at 0 s, (b) at 181 s, (c) at 344s, (d) at 591 s. The scale bar represents 1 µm. Reprinted from Ref. 146. Copyright (2001),with permission from the American Chemical Society.


potential value. Similarly, advances in instrumentation in other techniques suchas NMR, Raman microscopy, and electron microscopy toward higher spatial andtemporal resolution, as well as new methods for materials synthesis that canproduce cleaner, better-defined polymers, promise to have a significant impact onour 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 evidencehas been gathered, but interpretation has often been problematic because of theambiguities 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, itis now coming under increasing pressure, from experimental data, from computersimulation, 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 adjustmentsto the theory, or whether the idea of a nucleation barrier is flawed, or even ifthe idea that the crystal thickness seen is the fastest growing is correct. Withthe development of new theoretical tools, and the increased integration of theorywith computer simulation, it is hoped that a more complete model for polymercrystallization can be developed.

A full understanding of crystallization from the melt still seems elusive, withvery 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 fracturethe growing crystal (156)], or of the relationship between modes of growth andmorphologies 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 isan area that, with the development of new experimental techniques, and the push-ing of the spatial resolution of existing techniques down toward the nanometerlevel, should be resolvable.

Despite these questions, the body of knowledge on crystallization kinetics isvery 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 ofdifferent chemistries and behaviors that are observed can lead to a temptation todiscard generalizing principles, but it is the author’s opinion that this temptationshould be avoided. At a phenomenological level, polymer crystallization is wellcharacterized, 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, newlyavailable experimental data can be married to the vast store of existing knowledgeand 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 placedon them by the emerging fields of nanotechnology and nanoscience.


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JAMIE K. HOBBS

University of Bristol

CURING. See THERMOSETS.

CYANOACRYLIC ESTER POLYMERS. See POLYCYANOACRYLATES.

CYCLIC OLEFIN POLYMERS.See ETHYLENE-NORBORNENE COPOLYMERS.

Crystallization Kinetics

Technology

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