Self-Generated Fields and Polymer Crystallization - Macromolecules

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Self-Generated Fields and Polymer Crystallization Jerold M. Schultz Department of Chemical Engineering and Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716, United States ABSTRACT: The creation of thermal, compositional, and stress fields during the crystallization of polymers from the melt is described. The treatment of crystallization under self-generated fields is reviewed, including classical moving boundary problems, the treatment of dendrite growth, and coupled growth. The extension of these treatments to polymer crystallization requires that the velocity of interface motion be defined by the temperature and composition of the melt at the solid−liquid interface, a feature not found in extant analyses suitable for small-molecule systems and metals. Inclusion of this feature renders analytical solutions difficult and usually requires the use of numerical methods. The role of the diffusion length in defining morphological features is described. Methods of simulating the growth of polymer spherulites are reviewed. These include finite element, analytical, and phase field approaches. The role of thermal fields in fiber processing is discussed. Finally, speculations regarding the role of stress fields are presented.

A. OVERVIEW: SELF-GENERATED FIELDS AND PATTERN FORMATION To a considerable extent, the properties of a material are established through control of structural, including morphological, detail. It is for this reason that a large body of literature relating to morphological development has emerged. The details of morphological development may differ from system to system. Nonetheless, for transformations occurring through a growth mechanism (as opposed to uniform transformation or to spinodal decomposition), certain elements are common: a driving force, microscopic attachment at the growth interface, dissipation of the heat of transformation, and, if necessary, transport of crystallizable and uncrystallizable molecular species to and from the interface. Flexible polymer molecules crystallized from a quiescent melt typically form thin, lamelliform crystals, a few nanometers to a few tens of nanometers in thickness. Further, these crystalline lamellae stack periodically, one above the other in the thin direction. The crystals are separated from each other by noncrystalline matter, this intervening layer having a thickness at the same dimensional level as do the crystals. The root cause for the formation of such thin crystals has been an active topic for over a half century and remains so today. While important and fascinating, this question lies outside the scope of the present review. Accepting that lamellar crystals are a basic result of polymer crystallization from the melt, we here ask about the higher level cooperative development of colonies of such crystals into stacks of lamellae, growth arms, and spherulites. Posited here is the concept that such higher-level morphological development arises as a natural result of fields created at the growth front. As we will see, the fields may be compositional, thermal, or mechanical (stress). The presence of such fields reduces the propagation velocity of the crystallization front. But the ability of the front to modify its morphology allows adjustment of the rate of dissipation of the field so that the growth velocity may be optimized according to some physical principle. © 2012 American Chemical Society

A well-known example of such behavior is the formation of snowflakes.1 Each snowflake is an ice crystal which has grown in the form of thin arms and branches, as shown in the micrograph of Figure 1. The hexagonally organized arms display the

Figure 1. Photograph of a snowflake, from Kenneth G. Libbrecht, snowflake.com.

symmetry of the ice crystal. That the arms form at all and are quite thin is the result of a compositional field problem. The driving force for water crystallization here is very high. The driving force derives from the high supersaturation of the water vapor at temperatures below its equilibrium melting point Tm. Once a crystal of ice nucleates, it begins to grow as a hexagonal disk, as dictated by crystal symmetry. Because of the high driving force, the growth velocity of the crystal should be very large (see later), but the growth velocity is actually much smaller due to the low rate of arrival of randomly moving gaseous water molecules to the growth surface. If the crystal were to grow as a smoothsided hexagonal plate, molecules would approach the growing Received: November 9, 2011 Revised: April 12, 2012 Published: June 25, 2012 6299

dx.doi.org/10.1021/ma202476t | Macromolecules 2012, 45, 6299−6323

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surfaces of the plate only from the vapor directly in front of the growing surfaces. By crystallizing in the form of very thin arms, the very narrow tips of the arms can accept the water molecules from the surroundings over a solid angle of nearly 4πmuch more efficiently than for a large flat surface or a sphere. Because of this increased rate at which randomly moving molecules approach the growth tip of the thin arm, the crystallization of the arm proceeds at a relatively high rate. Consequently, given that the formation of a narrow arm is possible, this is the geometry which nature would select. We shall see later that high driving forces create a morphological instability which results in the creation of narrow arms. The process just described can be couched in terms of the growth of a crystal in a concentration f ield. As the water vapor molecules join the crystal, there is a large depletion of water vapor near the growth surface. Far from the growth surface the water concentration is much larger, and a smooth gradient of concentration therefore exists in the vapor from the surface outward; this describes a concentration field. The diffusion of molecules in a concentration field is the well-known Fickian physics (convection can also play a role). We can see now that the selected morphology of the growth surface is that which somehow couples the velocity of growth of the crystal to the rate of diffusion (or convection) of molecules through the concentration field to the growth surface. Effects similar to those which create snowflakes are seen when polymer single crystals are grown from solution. Here, as in the case of snow crystals, patterns develop in response to the difficulty in supplying polymer molecules to the growing crystal at rates fast enough to optimally affect growth. Examples are shown in Figures 2 and 3. Figure 2, taken from Reneker and

Figure 3. Transmission electron micrograph of a dendritic multilayer polyethylene crystal grown from ca. 0.1% xylene solution. Reproduced with permission from ref 3. Copyright 1961 Wiley.

more below the equilibrium melting point Tm, the driving force for crystallization increases and the velocity of the growth front must increase. Crystallization is an exothermal process; as crystallization occurs, the heat of fusion is liberated. Consequently, the temperature of the growth front should increase, and since the velocity of growth increases with undercooling, growth should become slower as crystallization continues and more and more heat is liberated. But again, when the growth surface can devolve into sharp asperities, the removal of heat through a three-dimensional thermal field becomes more efficient and more rapid, allowing crystallization of the narrow features to proceed rapidly, since the tips are in contact with melt at a relatively low temperature. Again it is expected that a morphology which couples the velocity of growth to the flow of heat through a thermal field will ensue. More specifically, when the flow of the heat of transformation from a planar or spherical interface cannot keep pace with the natural growth velocity of that interface, sharply pointed asperities must form in order to provide a higher dimensionality of heat flow and consequently more rapid heat dissipation. What can form is dendritic crystals (thermal dendrites): thin crystals with narrow branches of similar thickness. The lateral dimensions of such thermal dendrites decrease with increasing driving force (e.g., with increasing undercooling). Likewise, when nonincorporable molecules must flow away from the growing interface, cellular microstructures or “concentration dendrites” must form (which of these depending on the details). Heat and solute dissipation can have the effect of producing very fine and complex structures. The processes whereby such microstructures form has come to be known as “pattern formation”. To a significant extent, the physics of pattern formation is understood (see refs 4 and 5 for reviews), although general agreement on an optimization principle is lacking.

Figure 2. Polyoxymethylene crystals grown from dilute solution: left, transmission electron micrograph of star-shaped multilayer crystals; right, depiction of growth planes within the crystal. Reproduced with permission from ref 2. Copyright 1960 American Institute of Physics.

Geil,2 shows a multilayer polyoxymethylene (POM) crystal grown from dilute solution. The right side depicts the growth planes (fold planes), the red hexagon denoting the regular shape of a more slowly grown crystal. The sketch depicts new growth planes emanating from the corners of the crystal. It is these corners that protrude farthest into the solution and consequently can gather POM molecules from a greater angle (or volume of solution) than can sites farther from the corners. Consequently, the corners can grow more rapidly than can the rest of the growth face. Geil and Reneker suggested this effect in 1961.3 Figure 3 shows a more extreme deviation from regularly faceted crystals. This figure shows a dendritic crystal of polyethylene, grown from a ca. 0.1% xylene solution.3 In addition to concentration effects, thermal fields can come into play. As the temperature T of a melt is reduced more and

B. MORPHOLOGY OF POLYMERS CRYSTALLIZED FROM THE MELT The morphology of polymer crystals, whether formed from the melt or from solution, is that of lamellae or laths which are of the order of 10 nm thick in the chain direction. When crystallized from the melt, these crystallites organize into growth arms 6300


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and ultimate morphology of the growth arms and the details of redistribution of noncrystallizable matter fall naturally into the domain of pattern formation analysis. Three-dimensional spherulites, which are typical, can form only if the orientation of crystals about the radial direction (the growth direction) varies during the growth process. Any mechanism that creates crystal stacks with new stacking directions can produce three-dimensional spherulites. However, under a wide range of crystallization conditions, this variation in orientation is cooperative and regular, perceived as a twisting of crystals about the radial direction. Such crystal twisting is seen in a wide variety of inorganic and organic small-molecule systems as well as in polymers. It is possible that such twisting is also a response to a self-generated field.17

(stacks of laths, the stacking direction being the thin dimension), and the growth arms, in turn, organize into spherically symmetric bodies, called spherulites. Crystals within the stacks are separated by noncrystalline (amorphous) layers, whose thicknesses are usually of the same order as those of the crystallites. This hierarchy is depicted in Figure 4. (In this figure, the thickness of

C. SELF-GENERATED FIELDS Three kinds of fields can be self-generated during the crystallization of polymers from the melt. These fields are compositional, thermal, and stress. Compositional effects occur when (a) the melt (or solution) is a mixture of two or more components and (b) the composition of the growing crystals is different from that of the melt. For polymers, the exclusion of a chain could be absolute, as for molecular chains whose structure (chain chemistry or defect level) renders them thermodynamically unstable in the crystal, or kinetic, as for chains which could be incorporated in the crystal but whose kinetics of incorporation exclude them at high growth velocities. The kinetic exclusion refers to chains whose molecular mass or defect level produce slow kinetics of attachment. The components rejected by the growing crystal must migrate away from the propagating crystal/ melt interface. Generally, the rate of migration of the noncrystallizing species from the interface is slow enough that high concentrations of the noncrystallizing moieties remain in the melt near the interface, with the concentration decreasing with distance to the far-field level. Crystallization is an exothermic process, and the heat of fusion must be deposited at the growth front. If crystal growth is rapid relative to the rate at which the heat can flow away, a thermal field is created, with high temperatures in the melt near the growth front and decreasing with distance from the growth front. The elevated temperature at the growth front lowers the driving force for crystallization and consequently lowers the growth velocity (see eq 2). Under conditions of quiescent crystallization, the thermal diffusivity is high enough and the growth velocity slow enough that dissipation of this heat into the melt is almost instantaneous and the temperature of the melt at the growth front is little changed from the far-field value. Thus, the thermal field normally poses no hindrance to motion of the growth front. However, if the natural velocity of crystal growth is sufficiently high, thermal diffusion could also play a major role in pattern formation.18 Such conditions of extreme driving force are expected in the crystallization of highly oriented material. This type of transformation is expected in high-speed spinning, in postspinning heat treatments of partially oriented yarns, and under some conditions of blown film formation. In such cases of rapid, oriented crystallization, transmission electron microscopy of fibers19,20 and films21−27 has revealed very fine fibrillar crystals, the fibril axis coinciding with the draw direction. The fineness of the fibrils is determined by heat flow. Stress fields also form during the crystallization of polymers from the melt. Such stresses result from the high viscosity of polymer melts. The forming crystalline phase has a lower specific volume than does the melt from which it forms. Consequently,

Figure 4. Depiction of a growing spherulite, showing crystalline lamellae and growth arms.

the crystals is deliberately drawn smaller than that of the amorphous layer, to more clearly depict the amorphous layer.) In order to create such stacks of crystals, a mechanism must exist by which an existing growing crystal spawns similar crystals stacked below and above itself. Growth about giant screw dislocations screw dislocations with Burgers’ vectors along the chain axis and equal in magnitude to the crystal thicknessis observed (see ref 6 for a brief review) and is believed to be the agent for the creation of stacks of lamellar crystals. The constituent crystals are frequently (but not always) observed to twist about the radial directions of the spherulites. Kinetic details of the growth of ribbonlike or lamelliform crystals, without consideration of dissipative processes at the interface, are contained in molecular theories of attachment at the crystal/melt interface. As such, these attachment kinetics are included in expressions for and measurements of the natural growth velocity Vn. The interleaved stacks of crystalline and noncrystalline matter have a characteristic spacing of the order of 10 nm. Such lamellar crystallites appear to be a fundamental growth form, always present within spherulites and with thicknesses reflecting the polymer type and the temperature and pressure of crystallization. At this level, only the interlamellar region is highly affected by noncrystallizable components, these regions acting as “sinks” for at least a portion of the noncrystallizable material, as indicated sometimes (see later) by an increase in the thickness of the interlamellar region as the concentration of noncrystallizable species increases.7,8 The bundles of crystallites or growth arms (also sometimes referred to as f ibrils within spherulites) can be seen under optical microscopy and under atomic force microscopy. They are most clearly visible during crystallization, when the growth arms have not yet impinged on each other and grown together. Growth arms may or may not form, depending on the nature and concentration of any noncrystallizable matter which had been present in the melt and on the solidification temperature. In general, the arm thickness and spacing become finer with decreasing crystallization temperature.9 It is accepted that arm thickness and spacing are at least partially defined by the redistribution of noncrystallizable material during crystallization.7,10−15 Associated with this same question is that of the final location of domains of noncrystallizable material.10,16 The genesis 6301


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where Qt is the activation energy for segmental transport across the crystal/melt interface, T is the temperature at the crystal− melt interface, T1 is a temperature some 30 °C below the glass temperature, ΔT = Tm − T is the undercooling, and the driving force is

we may think of the crystal surface as shrinking away from the surrounding melt during crystallization. Were the viscosity very low, the melt would simply flow to level its density. If the viscosity is relatively high, there would be created a gradient of melt density from a low value near the growth front to a higher far-field level. This density gradient is produced from a highly entangled melt and is experienced as a stress field (negative pressure), highest at the growth front and decreasing away from the front. The index of importance here is the relaxation time of the melt. If this is sufficiently high, then the stress field will build as the growth front propagates, and the negative pressure would lower the melting point and suppress the growth velocity. That self-generated compositional fields must be important in polymer crystallization has been recognized for nearly 50 years, when Keith and Padden introduced the concept into the polymer literature.9−13 Almost all commercial semicrystalline polymeric materials contain some quantity of chains which are unincorporable in a crystalline phase due to chemical, steric, or molecular mass differences. Keith and Padden have provided a treatment of the effects of chain length effects on the concentration gradient in the melt near the growing interface.13 The same authors have produced a simplified model for the effect of growth surface geometry on spherulite growth. However, this latter analysis is not sufficiently detailed to provide a framework for a quantitative theory of pattern formation in polymeric systems. Goldenfeld15 has demonstrated the role of such “solute” flow in polymer crystallization using more recently developed elements of pattern formation. It can likewise be shown that heat flow must be very important during very rapid crystallizationas in high-speed spinlines. However, the existing analysis is rudimentary. While it has been recognized for some time that self-generated fields (SGF) are important in pattern formation and microstructure in commercial homopolymers,28 blends,16 and fibers,18,29 until recently there has been little in the way of predictive and quantitative theory specifically adapted to polymers. The situation is exacerbated by a paucity of growth rate, morphological, and diffusion data all in the same system, and these are the data needed to quantitatively test pattern formation models. It is an aim of this Perspective to explore those situations in which quantitative predictions can be made and compared with measurement.

ΔΦ = kK g /ΔT = abσSσeTm/LΔT

where a is a constant having value 2 or 4, depending upon the mechanism of growth (regimes 1 or 2),35 b is the interplanar spacing parallel to the growth direction, σS and σe are the surface energies of faces respectively parallel and normal to the chain axis, Tm is the melting point, k is Boltzmann’s constant, and L is the heat of fusion. If noncrystallizable chains, excluded from the crystal, build up in the melt at the growth front to a concentration cint B , this concentration of “solute” molecules can affect the natural velocity in several ways: sometimes by changing the mobility (changing Qt and T1), sometimes by altering the equilibrium melting point, and always by diluting the crystallizable chains at the growth front by a factor 1 − cint B . Incorporating only the dilution effect into (3), we can write Vn = A(1 − c B) exp[−Q t /R(T − T1)] exp(−K g /T ΔT ) (4)

While this expression does not include effects on the melting point, the heat of fusion, and the mobility, it is a good approximation for blends of molecules which are chemically the same or very similar but differ only in the tacticity or defect level of the mers, as, for example in isotactic/atactic mixtures.

E. CLASSICAL TREATMENT OF INTERFACE MOTION WITH SELF-GENERATED FIELDS E.1. Classical Analyses of Moving Boundary Problems. When the heat of fusion is liberated at the front of a growing crystal, there must be a coupling of the growth velocity and the diffusion of heat away from the interface. To obtain the point-bypoint temperature (or concentration) in the melt or solution ahead of the front, one solves the diffusion equation with a front propagating at a velocity V. In one dimension, this is termed the “directional growth equation”.36 In three dimensions this is written

D. LAWS OF INTERFACE MOVEMENT: THE NATURAL GROWTH VELOCITY In the small molecule literature, the science of phase transformations and morphological control is highly developed. Processes occurring at the interface are well-known, and the kinetics have been modeled, to the extent that, for pure materials, laws relating to the effect of driving force on interface velocity are accepted. In general, these laws predict that, at low driving forces (i.e., near the equilibrium melting point), the interface velocity V, referred to here as the “natural velocity”, is a low power (p = 1 or 2) of the driving force ΔΦ Vn = Mc(ΔΦ) p

DT ∇2 T + V ∇T = 0

(5)

and D∇2 c Bm + V ∇c Bm = 0

(6)

where DT and D are respectively the thermal and mass diffusivities. Equations 5 and 6 represent the conservation of energy or mass in a volume element: what flows in due to the movement of the front (second term) must equal what diffuses out (first term). The classical case is that solved by Stefan in 1891.37 Stefan considered the melting of an ice sheet. In this case, melting occurs from the top down, as a planar front. Since superheating is very difficult to achieve, the moving front must always be at the melting point of ice. The situation is as depicted on the left side of Figure 5. Here, as the solid/liquid front moves downward, the temperature of the front is fixed, but the gradient of temperature can change as the front moves. Since the temperature of the front is constant, heat must be liberated at the front at exactly the same rate at which it is conducted away. The flux of heat liberated at the front is VρL/M (cal/(cm2 s)), where L is the heat of fusion,

(1)

where Mc is a constant. Implied here is that if the driving force is changed (e.g., by an increase of temperature at the growth front), the growth velocity changes accordingly. For polymers undergoing quiescent crystallization, widely accepted empirical forms of the interfacial attachment kinetics are likewise available.30−34 Here the relationship between Vn and the temperature of crystallization is of the form Vn = A exp[−Q t /R(T − T1)] exp(−K g /T ΔT )

(3)

(2) 6302


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where erfc(z) is the complementary error function ercf(z) = 1 − erf(z) and erf(z) = (2/π 1/2)

∫0

z

2

e −ξ dξ

and 2

A = λe λ Lπ 1/2/cp

λ being the root of Figure 5. Moving boundary problems: left, the Stefan problem of a crystallizing (or melting) ice sheet; right, polymer crystallization.

Vρ L = −DT cpρ∇T Q̇ = (7) M where Q̇ is the flux of heat across the interface. Similarly, when the concentration csB of component B in the solid is lower than that in the melt (cm B ), the flux of excluded B must equal the rate at which it diffuses away:

csB

−

Dcpρ dT

L ρ dX Lρ V = M dt M

=

M dx

(10)

where cp is the specific heat of the melt, ρ is the mass density of the melt, M is the molecxular weight, and V is the velocity of interface propagation (commonly referred to as the growth rate). Inserting (9) into (10) gives, after some algebra

(8)

cm B

where and are the concentrations of B in the solid and the melt, respectively, and D is the mass diffusivity. Equations 7 and 8 relate the velocity of growth to the thermal or compositional gradients. If (5) and (7) (or (6) and (8)) can be solved simultaneously, one obtains the velocity of the front at each position of the moving front (or each moment in time). If the velocity V may take any value, an analytical solution is possible. For small molecules, this is typically true; in this case, there is little energetic barrier to the incorporation of new molecules to a growing crystal, and growth can therefore occur at a very small driving force (very small undercooling). In this case, we would say that the Stefan problem is solved. Such “moving boundary” problems are classical situations in diffusion theory (see e.g. Carslaw and Jaeger38). Two standard cases are those of (a) a planar crystal/melt boundary and (b) a spherical crystal/melt boundary moving into a semi-infinite undercooled melt initially at T0 < Tm, where Tm is the melting point of the solid. We shall consider those solutions momentarily, as they are instructive. It is important to note at this point that for polymeric systems the energetic penalty (both enthalpic and entropic) is relative large. Analytical models for the incorporation of polymeric strands to a growing crystal are generally expressed in terms of a nucleation event and require significant undercooling to proceed at measurable rates. The velocity is not free to take any value but is dictated by the temperature of the melt and the local concentration of uncrystallizable species at the interface, as in eq 4. For this case, the velocity of the front is fixed by the temperature and concentration of the melt at the front, but those values depend on the solution to the diffusion problem. There is no known analytical solution for this situation. Returning to the Stefan problem, the simplest case is that for a planar crystal/melt boundary growing into a supercooled melt of melting point Tm, far-field (initial) temperature T0, heat of fusion L, thermal diffusivity DT, and heat capacity cp. The solution for this case is ⎛ x ⎞ ⎟⎟ T (x , t ) = T0 + A erfc⎜⎜ ⎝ 2 DT t ⎠

πL

Numerically, λ is found to be of order unity. From (9) we see that the temperature of the melt decays from Tm at the crystal/melt interface to the far-field value, T0. The rate at which heat of fusion is liberated at the interface must match the rate at which it diffuses away, giving the following boundary condition at the interface position X:

M the molar mass, and ρ the mass density. The flux of heat diffusing into the melt is −(DTcpρ/M)∇T, where ∇T is evaluated at the interface. The balance is thus

V (c Bs − c Bm) = −D∇c Bm

(Tm − T0)cp

2

λe λ erfc λ =

X = 2λ DT t

(11)

and V = λD1/2t −1/2

(12)

Thus, the growth must decelerate according to a parabolic law. What is true for the thermal field must also be true for a compositional field; a parabolic dependence on time is expected for a growing planar interface which rejects an unincorporable species as it propagates, this unincorporable species displacing the crystallizable species in the melt at the growth front. Of more interest to us is a growing sphere (think “spherulite”). The solution of the spherical problem gives the temperature T(r,t) at time t and radius r ahead of the growing sphere of instantaneous radius R as T (r , t ) = T0 + − 2

e

−λ

⎧ (D t )1/2 2 ⎨ T e−r /4DT t r erfc λ ⎩

2λ(Tm − T0)

⎪

2

⎪

− λπ

1/2

⎫ r π2 ⎬ erfc 1/2 2 2(DT t ) ⎭ ⎪

⎪

(13)

2

where λ2eλ {eλ − λπ1/2 erfc λ} = 1/2clp(Tm − T0)/L and clp is the heat capacity of the melt. If we wish to follow the position of a specific temperature at increasing times t, we see from (13) that the lumped variable ((DTt)/(r2)) must be constant for all time t. Thus, the position r for temperature T must go as r = (const) × DT 1/2t 1/2

(14)

Analogously to (10), the matching of thermal fluxes at the interface is −

DT cpρ dT M

dr

=

L ρ dR M dt

(15)

where R is the radius of the growing sphere. Using (13) in (15) and performing the necessary algebra, we find

(9) 6303


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R = λ DT t

δ ≈ DT/V from the boundary. This distance δ is a measure of the extent of the thermal field created by the moving boundary and is termed the dif f usion distance. A very similar result is found for most other geometries and boundary conditions. And of course the same result holds for a compositional field. Consequently, δ = DT/V is used as a universal measure of the extent of the thermal field in front of a moving source of heat. In exactly the same manner, the extent of the compositional field in front of a moving “source” of matter is D/V, where D is now the molecular diffusivity of the material. The crystallization of a melt for which the composition of the crystal is lower in noncrystallizing component B than that of the melt is such a situation. In this case, the propagation of the crystal-melt front into the melt is the effective “source” of an excess concentration of B in the melt. While the diffusion distance, as a measure of how far heat or matter will diffuse ahead of a moving interface, is a useful quantity, it is often even more useful to know how that distance compares with the distance heat or matter must diffuse in order to promote some process. If Δ is the distance that heat or matter must diffuse to some specific end, then it is the ratio Δ/δ that becomes important. This ratio, Δ/δ, is known as the Peclet number for crystallization: Pe = Δ/δ. Consider the morphology of polymer crystallization from the melt: stacks of lamellar crystals forming growth arms. If one is interested in the interaction of thermal or compositional fields between adjacent crystal lamellae or adjacent growth arms, the appropriate Peclet numbers would be L/δ and LGA/δ, respectively, where L is the periodicity of the lamellar stacking and LGA is the interarm spacing. If L/δ (for crystal lamellae) is significantly greater than 1, then the thermal or compositional fields of adjacent lamellae do not significantly overlap and the crystals should grow as if there were no neighbors. If L/δ is significantly less than 1, then the fields strongly overlap. Because of the overlap of concentration fields, the concentration of excluded species or the temperature in the melt near the interface is increased, and the growth velocity must decrease, through a dilution effect or through a reduced undercooling. Likewise, whether LGA/δ is greater or less than 1 dictates the degree of overlap of the fields of adjacent growth arms. When the growth velocity is impacted by the overlap of fields, the system usually restores its growth velocity, at least partially, by altering its morphology so as to increase the ratio Δ/δ. For instance, this can be done by increasing the pertinent spacing Δ. This is perhaps the most important concept to be learned here. Let us consider crystallization of component A from a meltmiscible A/B blend. Here we shall refer to the B component as the “solute”. One can immediately distinguish four cases: (1) L/δ significantly greater than 1 (Pe ≫ 1). This is sketched in Figure 6a. Here the solute fields are very localized and do not overlap at all. The crystal lamellae grow independently of each other. Similarly, the growth arms grow independently of each other, since LGA/δ must be even greater than L/δ. This condition can obtain at relatively large undercoolings, and the result would be no change in morphology, relative to the neat polymer, at either the interlameller or interarm level. The velocity of the crystallization front would, however, be lower than that of the neat polymer, due to solute buildup at the front. (2) L/δ < 1 and LGA/δ significantly greater than 1. In this case, the growth of lamellae within a growth arm is impacted, while there is no other effect on the individual growth arms. The expected result would be an increase in

(16)

Again, non-steady-state growth with parabolic kinetics is found. And the analogous situation holds for a crystallization front moving into a multicomponent melt or solution for which one or more of the components are rejected by the crystal and form a compositional field in the melt. While (16) cannot exactly hold for polymer crystallization (since we have seen that this is mathematically a somewhat different problem), it is widely assumed that these equations should approximately hold for polymer crystallization: the propagation of a planar or spherical front should exhibit parabolic kinetics. Equations 9 and 13 for the temperature in front of a growing plane or sphere show that that temperature increases as the interface moves forward (increasing time). At the same time, the temperature gradient into the melt decreases with time. This can be seen by looking at dT/dx or dT/dr. The derivative of erf(z) 2 with respect to z is just (d erf(z))/dz = e−z So for the planar case, using (9) dT dx

X

⎛ A −λ2⎞ −1/2 =⎜ e ⎟t ⎝2 D ⎠

(17)

telling us that the temperature gradient decreases as the interface moves forward. From the matching condition at the interface (eq 10), this decreasing temperature gradient demands a correspondingly decreasing interface velocity (as we have already seen). An exactly analogous set of equations pertains to compositional fields. E.2. Diffusion Length and the Peclet Number for Crystallization. The diffusion length is the distance over which the temperature, or the concentration of uncrystallizable species, drops to a specified level. The diffusion length turns out to be a convenient parameter, as we shall see later. Using (3), for the propagation of a planar front, we can write for the ratio of the temperature at a distance x′ from the origin to the temperature at the interface erfc T (x′, t ) − T0 = T (X , t ) − T0 erfc

x′ 2 DT t X 2 DT t

(18)

Given (11), the denominator can be written as erfc λ. For a specific ratio B for the left side of (18), we have then B erf λ = erfc

x′ 2 DT t

For B = 1/e and λ taken as 1, and representing the distance to obtain the relative temperature drop of 1/e as δ, we have erfc λ

δ = 0.368 2 DT t

Giving

δ = 1.28 DT t

(19)

Differentiating eq 16 with respect to time, we have V = λ/2(DT/t)1/2. Using this, we substitute for t in (19):

δ = 0.64λ

DT D ≈ T V V

(20)

Equation 20 shows that the temperature ahead of the planar boundary falls to 1/e of its value at the boundary at a distance of 6304


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micrographs of spherulites grown to completion in a 20/ 80 poly(butylene succinate)/poly(butylene adipate) (PBS/PBA) blend in which the PBS has crystallized isothermally at (left) 75 °C and (right) 100 °C. For this system, the melting temperatures of PBS and PBA are respectively 114 and 60 °C. So, in both cases of Figure 7 PBS should crystallize at the temperature of isothermal crystallization, and the PBA should crystallize upon subsequent cooling to room temperature. Here the diffusion length at the higher crystallization temperature is ∼300 times larger than that at the lower temperature. In these micrographs, the (a) micrographs were taken at the crystallization temperature, at which the PBA is still molten, while the (b) micrographs were taken at room temperature, after the PBA has also crystallized. One sees here that at the higher crystallization temperature the PBA has been excluded from the PBS spherulites, while at the lower crystallization temperature, the PBA crystals are fully incorporated within the PBS spherulites, both as expected. Figure 8 shows the growth kinetics at 100 and 75 °C for this blend. As expected, parabolic growth is seen for crystallization at 100 °C, while a constant growth velocity is seen at 75 °C, where the diffusion length is much smaller and the solute collects between the growth arms. An analogous result has been reported for the poly(vinylidene fluoride)/poly(butylene succinate) system. 40 Keith and Padden11 and Okada et al.41 likewise report a transition from linear to parabolic growth kinetics as the diffusion length of blends is increased. E.3. Two- and Three-Dimensional Problems: Dendrites. We have seen that in the cases of planar or spherical growth the

Figure 6. Depiction of the extent of overlap of fields. The arcs show diffusion length distance from the centers of the tips of crystalline lamellae: (a) small diffusion length; (b) large diffusion length.

interlamellar spacing, with no change in the interarm spacing. We shall see in section F.2.2 that this is borne out. (3) L/δ ≪ 1 and LGA/δ near 1. This situation is sketched in Figure 6b. For this situation, possible at relatively small undercooling, lamellar stacking would be unaffected, since the solute would diffuse beyond the interlamellar spacing. But in this regime the growth arm spacing should increase, to accommodate the accumulating solute. (4) L/δ ≪ 1 and LGA/δ ≪ 1. In this case, the solute fields beyond the growth arms are strongly overlapped, and consequently diffusion is possible only parallel to the growth direction. This is the case of a growing sphere, which we have already seen. Here the solute would be largely excluded from the spherulite and the growth velocity would be parabolic. An example of this behavior is shown in Figures 7 and 8.39 Figure 7 shows optical

Figure 7. Polarized light micrographs of a PBS/PBA 20/80 blend in which the PBS crystallized isothermally at (left) 75 °C and (right) 100 °C. Upper micrographs were taken at the PBS crystallization temperature; the lower at room temperature, following the crystallization of PBA. Reproduced with permission from ref 39. Copyright 2008 Elsevier. 6305


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systems grow at a constant velocity. In all such cases, the growing crystals are very thin and growth is via the propagation of a fine tip or fine edge. The simplest case of growth with a fine tip is that of dendrites. Snowflakes (as in Figure 1a) are an example of dendritic growth. Snowflakes form in response to compositional fields. The snowflake is composed of six primary arms symmetrically arranged in a plane, with each arm exhibiting secondary branches and, sometimes, tertiary branches. Each arm with its branches is a dendrite. Dendritic growth refers to any treelike growth pattern and is relatively commonly found in nature (and technology). Examples of dendrites crystallized from melts of a binary metallic alloy42 and a polymer blend43 are shown in Figure 9. A dendritic polyethylene crystal grown from a dilute xylene solution is shown in Figure 3. What is important to observe in all dendrites is that the arms are thin rods or plates and are sharply pointed. In all of the cases just cited, the fineness of the structural moiety is a response to the difficulty of the flow of heat or matter at the growth front to keep up with the propagation of that front. For the case of a rodlike dendrite, the fine propagating tip allows heat or matter to diffuse over essentially 4π solid angle, as opposed to unidirectional diffusion of heat from a propagating plane or spherical surface. This geometrical effect is quite efficient and permits coupled interface propagation and heat or mass diffusion at rates sometimes orders of magnitude faster than for the case of a plane or sphere. It is not accidental that polymer spherulites also grow via the propagation of fine crystalline fibrils. Saratovkin, in the early 1950s, was the first to recognize the conceptual physics of dendritic growth and provided qualitatively correct explanations for the phenomenon.44 Saratovkin’s qualitative explanations were fleshed out, beginning in the late 1950s, by quantitative, predictive theories. Below is given a brief and somewhat simplistic summary of the treatment of dendritic growth, the purpose being to provide a framework of understanding for the polymer work reviewed later. Analytical treatments of the unconstrained growth of an isolated rod of constant tip geometry growing into an infinite melt held at a temperature T0 (lower than the melting point Tm) were first published in the early 1950s. The lengthwise growth of thin rods is a steady-state (constant velocity) problem and has been solved analytically for paraboloids of circular45 and of elliptical46 cross section. As usually constituted, the problem is set up as a moving boundary problem of diffusion (see ref 38, for moving boundary analyses). The Stefan problem, described

Figure 8. Radius of PBS spherulites versus crystallization time for PBS/ PBA 80/20 blend crystallizing at (a) 100 °C and (b) 75 °C. The line in (a) is the least-squares parabolic fit, and the line in (b) is the leastsquares linear fit. Reproduced with permission from ref 39. Copyright 2008 Elsevier.

diffusion of heat or molecules to or from the growth front is onedimensional, normal to the growth front. In this case the heat or uncrystallizable molecules continuously build in the melt near the growth front and the growth velocity always slows, according to a t−1/2 law. It is remarkable, however, that some physical

Figure 9. Dendrites in materials crystallized from the melt: (a) X-ray microradiograph of tin-rich dendrites growing in a Sn−13% Bi alloy. Reproduced with permission from ref 42. Copyright 2004 American Physical Society. (b) Poly(ethylene succinate) dendrites growing in the melt of a miscible poly(ethylene succinate)/poly(ethylene oxide) blend at 65 °C. Reproduced with permission from ref 43. Copyright 2012 John Wiley & Sons, Inc. 6306


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where y = Vρ/2DT. While (25) can be solved to obtain a relationship between V and ρ, another principle must be invoked to establish absolute values. The relative values of several quantities are important in the evaluation of (25). Of first importance are the four lengths: ρ, the tip radius; DT/V, the thermal diffusion length; Ds/V, the “solute” diffusion length; γs/L, the capillary length. The tip radius is generally of the order of magnitude of one of the diffusion lengths; consequently, these are indices of whether or not the interface will be stable. The value of the capillary length acts to set the absolute value of ρ. Additionally, the “kinetic undercooling” δT(V), which establishes a natural interface velocity consistent with the interface temperature, is important. Finally, convective effects, not treated in the analysis leading to (25), can also act to set the thermal field near the interface. Temkin,52 in 1960, and Bolling and Tiller,47 in 1961, produced the first attempts to define absolute values of the growth velocity and tip radius, Temkin invoking a condition of maximum growth velocity and Bolling/Tiller testing both maximum velocity and maximum rate of entropy production as optimization principles. The latter authors concluded that maximum growth velocity provided the better fit with experimental results. Bolling/Tiler also pointed out that the assumption of an isothermal interface is inconsistent with steady-state growth and gave the first analytical treatment of growth with a nonisothermal interface. Finally, these authors pointed out, correctly, that the temperature of the melt at the interface must be lower than the equilibrium melting point, even when adjusted for capillarity, since finite melt undercooling is necessary to match the natural interface velocity Vn (see eq 4) to the velocity dictated by heat or solute dissipation. Shortly after the work of Bolling and Tiller, Mullins and Sekerka53,54 and Sekerka,55−57 working independently in the same laboratory as Bolling/Tiller, proposed a criterion for the stability of an interface toward wavelike modulations. In this case, surface energy and positive thermal or compositional gradients act to stabilize the interface, while negative thermal gradients and the solute buildup due to finite diffusivity are destabilizing. Their model permits the determination of the minimum value of V at which the interface will become unstable. Caroli et al. have reexamined interface stability, using a more general nonequilibrium thermodynamics model, at the same time releasing constraints of infinitesimal fluctuations, incorporating convection, and investigating meniscus effects which are present in microscope studies.58−62 Mullins and Sekerka’s contribution has been crucial in three ways: First is its intrinsic worth in providing a conceptually correct attack to the question of surface shape stability. Second, it provided a starting point from which the question of how the fields of adjacent asperities may interact. This is an important question which is not treated in the original dendrite work. Lastly, the concept of intrinsic instability of a nonplanar surface became recognized63 as important in determining the shape of an independently growing dendrite. In this last respect, Langer and Müller-Krumbhaar64,65 recognized that for a dendrite tip growing at some velocity the tip must become unstable for all tip radii above some critical value. On the other hand, decreasing the tip radius below the stability limit induces side branching, effectively increasing the radius back toward the stability limit. They then suggested that the operating condition must be exactly at the stability limit (“marginal stability”). This principle, in conjunction with velocity-radius predictions from the ellipsoid models cited above, appears to match experimental results in several systems.4,66−70

above, is the classical moving boundary problem. But now one seeks the steady-state condition for which the rate at which heat is evolved and/or solute is rejected at the growth front exactly matches the rate at which it diffuses away from the interface. For a fine dendritic rod, the rate of heat evolution and/or solute rejection is again specified by the growth velocity. The rate at which diffusion of heat or solute diffusion from the growth front occurs depends on the radius of the dendrite tip; the flow would be one-dimensional for a tip of infinite radius and would be fully three-dimensional for an infinitesimally small radius. Thus, the tip of the growing dendrite should adjust itself to whatever radius provides dissipation of heat or matter at the necessary rate. In the extant treatments of rodlike growth, no conditions are placed on the magnitude of the growth velocity; it is allowed any value which provides the match between evolution of heat or solute and its dissipation. The steady-state solution provides a relationship between dendrite tip radius ρ and growth velocity V, but some other principle must be invoked to establish absolute values of V and ρ. Among the assumptions inhering to early analyses of growth of a fine rod have been (1) an assumed interface shape, (2) an isothermal interface, (3) isotropy of surface energy and natural growth velocity, and (4) the absence of convection. The constraints of interface isothermality,47,48 given interface shape,46 and of nonconvectiveness49−51 have been treated, and to some extent removed, in subsequent work. Only in the more recent numerical simulations has the assumption of surface energy isotropy been relaxed. It is useful to review the classical, isolated dendrite model of Ivantsov.45 This solution will be used in section G, Pattern Formation during Fiber Spinning. For the growth of a narrow rod with an isothermal tip of circular cross section, growing in the presence of a thermal field, Ivantsov derived the following relationship between constant growth velocity V and tip radius ρ: −(Vρ /2DT ) exp(Vρ /2DT )Ei( −Vρ /2DT ) = (cp/L)[T (int) − T0]

(21)

where T(int) is the temperature of the noncrystalline phase at the growth surface and Ei (y) is the exponential integral function Ei(y) =

∫x

∞

(e−a /a) da

(22)

An analogous relation holds for growth into a compositional field, only the right side of (21) changing. (21) can now be solved for the product Vρ. In order to solve for the product Vρ, a value of T(int) must be obtained. For a pure substance T (int) − Tm[1 − (2γs/L)/ρ] − δT

(23)

where the term containing ρ and γs accounts for the melting point depression due to capillarity at the tip and δT accounts for the relation between undercooling and driving force. In general, the natural velocity is some function of the driving force, or, inverting this concept δT = δT (V )

(24)

Assuming the dendrite to be isothermal and inserting (23) and (24) into (21) −yey Ei( −y) = (cpTs/L)/[1 − T0/Ts − (2γs/L)/ρ − δT (V )/Ts]

(25) 6307


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Numerical methods have been used to simulate dendritic growth. Ben-Jacob et al. proposed a boundary layer method,71−73 and Brower et al. proposed a “geometrical” method74 to simplify numerical computations of pattern formation. In both cases, the gradient in temperature or concentration is confined to a narrow layer about the interface. This implies that the diffusion length is much smaller than the tip radius and restricts accurate computation only to the vicinity of the tip. By simplifying the numerical task, both groups were able to pose otherwise more realistic problems, notably including anisotropy in the surface energy71−75 and the driving force for crystallization.75−77 In both treatments, it was concluded that anisotropy is required for the growth of a stable dendrite tip. Goldenfeld has shown that below a critical level of anisotropy the tip continuously bifurcates, generating a “dense branching” microstructure,78,79 a morphology which resembles cauliflower. Polymer spherulites occasionally display this morphology. Goldenfeld has further shown that under those conditions under which a stable tip can grow, the operating condition is that of maximum velocity, and not that of marginal stability.78−80 This result of Goldenfeld suggests that the original models of Temkin and of Bolling and Tiller may yet have validity over some range of conditions. It is useful to point out here that there are two often-run types of dendritic growth experiments: free growth and velocitycontrolled growth. In the former, a specimen is quenched to the temperature T0 of an isothermal environment The environment is maintained at T0 and the growth velocity and morphological features examined during the free growth of the crystals.51,81−86 In the latter, the specimen is pulled through a stationary temperature gradient.67−70,82,87−90 In this case, also termed “constrained growth”, the growth velocity is fixed at the pulling velocity, and the environmental temperature and morphological features can be measured. While the optimization principle for coupled dendritic growth remains moot, Laxmanan’s90 correlations of data on constrained growth with various optimization conditions seem to indicate that minimum undercooling is governing in this case (at least for succinonitrile systems). Goldenfeld’s simulations of isolated dendrite growth78,79 agree broadly with this criterion. Simplistically, this can be envisioned as follows. Consider the melt directly in front of a solid−liquid interface moving toward lower temperatures in a fixed temperature gradient. At each sequentially lower temperature the interface tries to move at the imposed velocity, but a specific tip sharpness is required, to ensure that diffusion of heat or solute from the dendrite tip can match the growth velocity. However, as the tip sharpens, it also lowers the undercooling, through increase of its surface energy. If the tip were less sharp, the rate of diffusion would not keep up with the growth velocity, and the buildup of heat or solute near the interface would decrease the undercooling. A criterion for steady state would thus be that the tip radius would provide the minimum undercooling: dΔT /dρ = 0

Figure 10. Optical micrograph of a eutectoid steel partially transformed from austenite to pearlite (alternating platelets of iron-rich ferrite and iron carbide). The platelets are seen edge-on. The white regions are the unconverted regions of austenite, converted to martensite through quenching. Reproduced with permission from: Mangonon, P. L.; Oakwood, T. G.; Shapiro, J. M. In Metals Handbook, 8th ed.; American Society for Metals: Metals Park, OH, 1973; Vol. 8, p 188.

cementite (black, Fe3C) growing into the parent austenite (also white), which contains both iron and carbon. The fineness and mutual propinquity of the ferrite and austenite plates allow the transformation to take place much faster than would be the case if these grew as independent spheres or other shapes with massive surfaces. In such cases, the excluded component could diffuse away only parallel to the growth direction. We have seen that the competing kinetics of interface propagation and diffusion would then lead to a buildup of the excluded species near the growth surfaces and would slow and ultimately choke the transformation. Transformation in the form of a fine needle or plate allows diffusion normal to the growth direction and the possibility to obviate buildup of excluded species and to permit a constant growth velocity. The alternation of thin platelets provides a short path of migration for both species to enter the appropriate crystals; this path is normal to the growth direction. For the steel eutectoid of Figure 10, the carbon diffuses preferentially to the front of the carbide interface and the iron preferentially to the front of the ferrite platelets, as sketched in Figure 11. In eutectic transformations, alternating platelets of phases of different composition grow into a parent liquid. The growth of a polymer spherulite from the melt is an analogous situation. Here at one hierarchical level thin plates of essentially pure crystallizable polymer and intervening layers of crystallizable and uncrystallizable polymer grow into a parent melt. At a higher hierarchical level, crystalline growth arms and intervening layers of uncrystallized material grow simultaneously into the parent melt. Recognizing the similarity of these situations to the coupled growth of eutectoids and eutectics, it becomes useful to look in some detail at analyses of eutectoid and eutectic transformations. In a landmark publication in 1946, Zener provided an analytical model for the coupled, adjacent growth of solid phases α and β into the parent solid γ phase (a eutectoid transformation).91 Referring again to Figure 11, the alternating thin plates form as a response to the slow kinetics of diffusion in the γ phase of B excluded from a growing massive surface of α and A excluded from a massive surface of β. In those cases, the diffusion

(26)

For free growth, the situation is less clear, the possibility existing that maximum velocity controls in some systems, while marginal stability or some other condition controls in others, perhaps depending on the magnitude of the driving force. E.4. Coupled Growth. Eutectic and eutectoid transformations are examples of the coupled growth of thin plates of different composition. An example of a eutectoid transformation is shown in Figure 10. In this figure we see alternating thin platelets of iron-rich ferrite (white) and carbon-rich 6308


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Figure 11. Zener’s eutectoid transformation model. Left: alternating thin platelets of A-rich α and B-rich β grow into parent γ. Right: depiction of the diffusive motion of A and B during growth. Reproduced with permission from ref 91. Copyright 1946 the American Institute of Mining, Metallurgical and Petroleum Engineers.

F. TREATMENTS OF THE GROWTH OF POLYMER SPHERULITES F.1. Introduction. Three rather different methods have been used for quantitative treatment of pattern formation during spherulite growth. These are (1) finite element analysis, (2) analytical modeling, and (3) the phase-field approach. Each of these approaches has advantages and drawbacks, as we shall see in the following subsections. F.2. Numerical Simulation. F.2.1. Growth Arm Dimensions and Spacings. Consider, as an example, the syndiotactic/ atactic polystyrene spherulites shown in Figure 12. Here the noncrystalline atactic material (50 wt %) has been etched away, and one sees only the growth arms of the syndiotactic material. A finite element simulation modeling this situation has been carried out by Kit and Schultz,104 using input corresponding to the properties of the 50/50 syndiotactic/atactic polystyrene blends of Figure 12. The method, a three-dimensional simulation (in two spatial dimensions and time), is described in ref 105. The model simulates the case in which crystallization begins with one lamellar crystal. That crystal then spawns a new crystal, parallel and adjacent at each time period Δt, as depicted at the left in Figure 13. This is what would be expected for the spawning of new crystals by giant screw dislocations. The computational cell is shown at the right in Figure 13 as it would appear after a time 3Δt. With each increment in time, the diffusion equation is solved and the composition over all x, y found. Using the composition of the melt at the interface, the growth velocity is found for each crystal, using eq 4 and the interface stepped forward accordingly. The situation after 100Δt is shown in Figure 14, where the position X* of the growth front of each crystal is plotted against time (in number of steps). It is seen there that the crystals formed earliest grow rapidlyin fact, at a constant velocity. These are the crystals that have been able to dissipate the uncrystallizable chains laterally, as well as forward. The positions of the front after smaller amounts of time are also shown. The crystals which formed later in the process encounter a melt which has become enriched by the uncrystallizable chains rejected by the earlierformed crystals. The growth velocity of the later crystals is thereby reduced and indeed slows with time; conspicuously slower crystal growth becomes apparent around crystal number 10, terminating the steplike portion of the growth front, where diffusion is fast. This result is shown as Figure 15. Here are shown the front positions of lamelliform crystals which began growth at Δt = 1, 4, 8, 11, 40, and 60. The approximately parabolic growth of the crystals which formed at Δt = 11, 40, and 60 is due to the

would be one-dimensional and the transformation kinetics would be parabolic. By coupling the crystallizing of adjacent thin plates, the flow is rendered two-dimensional, and a constant growth velocity can be maintained. The principal diffusion direction of B is transverse to the growth direction, from in front of an α platelet, where it has been rejected, to in front of a β platelet, where it will be incorporated. Similarly, component A diffuses primarily laterally, from in front of β to in front of α. Zener’s contribution was to recognize the nature of this problem and then propose an analytical solution to this coupled diffusion− transformation process. Postulating that the system must grow at the maximum velocity consistent with the diffusion of the components, he showed that the period λ of the platelet stacking must be inversely proportional to the undercooling ΔT: λ ∝ 1/ΔT (where ΔT = Tm − T and T is the transformation temperature). Zener recognized that the diffusion problem vanishes as λ approaches 0; he also saw that the total α/β interfacial energy vanishes as λ approaches ∞. It is the effect of the creation of interfacial energy that requires a finite stacking periodicity. A maximum growth velocity exists between these two extremes. Hillert92 and Jackson and Hunt93 and subsequent investigators94−99 have refined Zener’s model, including the effects of interfacial curvature and spacing irregularity. Zener’s approach is likewise applied to the crystallization of eutectic mixtures from the melt.92,100 Another line of attack on coupled growth is to begin with the solution for a collection of isolated platelike dendrites and then allow for “solute” migration within the interarm or interplate volume. Most models are specific to constrained growth and have employed various simplifications and optimization principles. Burden and Hunt101 assumed (a) a spherical dendrite tip, (b) only longitudinal (i.e., parallel to the dendrite axis) concentrations in the interarm volume, (c) no direct effect of the thermal gradient, (d) no δT(V) term, and (e) minimum dendrite tip undercooling as the optimization principle. Later investigators have relaxed some of Burden and Hunt’s conditions or have used other conditions for steady growth.102,103 Very careful experiments have shown agreement with models based on a marginal stability condition67−69 or on a minimum undercooling condition.91 Somewhat later, finite element methods have been employed to simulate evolving interface patterns. The results of these computations indicate that growth conditions are less deterministic of cell wavelength than would be predicted from analytical models. In their present forms, none of the models can account for the prescribed natural growth rate for polymeric materials 6309


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Figure 12. Scanning electron micrographs of 50/50 syndiotactic/atactic polystyrene blends crystallized isothermally at different temperatures. In these specimens the atactic component has been leached away.

Figure 13. Geometry of the finite element simulation of the crystallization of a 50/50 sPS/aPS blend. Left: two (top and bottom) new lamelliform crystals begin growth every Δt time period. Right: the computational cell, showing one-half of the growing stack; lamellae seen edge-on. At each time increment Δt the diffusion equation is solved for the melt in front of each lamella and the concentration so found at the interface used to compute the velocity for the next step. Reproduced with permission from ref 104. Copyright 1998 John Wiley & Sons, Inc.

Figure 15. Computed position of the leading edge of lamellae 1, 4, 8, 11, 40, and 80 versus reduced time. Reproduced with permission from ref 104. Copyright 1998 John Wiley & Sons, Inc. Figure 14. Computed growth front after 20, 40, 60, 80, and 100 steps. Reproduced with permission from ref 104. Copyright 1998 John Wiley & Sons, Inc.

uniformly high concentration of uncrystallizable material at their growth surfaces. This high concentration is the result of lateral 6310


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We note that the composition of the crystalline lamellae is fixed at 100% component A, but the intervening layer is free to contain A as well as B. In this case, the interlamellar spacing L must vary in order to maintain the overall composition of the system. If cA is the overall mass fraction of A and fA is the mass fraction of A in the intervening layer, then the mass balance is expressed as

diffusion away from the earlier-formed crystals. This high concentration requires that, for crystals forming at times greater than Δt = 10, diffusion can occur only in the growth direction, the case for which parabolic growth is expected. Taking the breadth at half length of the protuberant arm in Figure 13 as the growth arm diameter, a comparison between simulation and measurement for the 50/50 syndiotactic/atactic polystyrene blends can be made. Order of magnitude agreement between simulation and experiment is found.118 It was mentioned above that the propagation of growth arms in spherulites is more often than not constant. Recall now that the concentration of uncrystallizable material (“solute”) in the melt adjacent to the growing arm (both longitudinally and transversely) starts at the as-mixed level and then must increase to the level associated with the steady-state propagation. This tells us that there must be a transient period over which the steady-state concentration field builds. This time to attain steady state depends on the diffusion length. Kit has performed a finite element simulation of this situation, using values of growth velocity and diffusivity appropriate to the isotactic/atactic blend system.106 The results show that when the diffusion length is small, the growth velocity comes to its steady value quickly but exhibits a long transient period when the diffusion length is large. F.2.2. Interlamellar Spacing. Consider now the possible effect of a noncrystallizable component B on L, the periodicity of stacking of crystalline lamellae of A within a spherulite. The situation is sketched in Figure 16. The view is edge-on to the

Lc L − Lc + fA = cA L L

(27)

giving

L = Lc

1 − fA cA − fA

(28)

If no A is present in the intervening layer, then L = Lc/cA, and this is then the smallest possible interlamellar spacing. So the periodicity L increases as more and more A is present in the intervening layer. The overlap of solute (uncrystallizable component B) fields in the melt are ameliorated by the increased distance between neighboring lamellae, thereby producing an increase in the crystallization velocity. The most rapid lamellar growth velocity must occur as L → ∞, since then the interference of neighboring fields is least. But then the mass transformed is vanishingly small. What seems a more reasonable scenario is that the rate of mass transformation is maximized. (The rate of mass transformation should be proportional to the rate of free energy decrease of the system.) This mass transformation rate is proportional to the product of the growth velocity and Lc/L, and this is the quantity which has been examined. The kinetics of this situation have been simulated using a finite element method. The computational cell is drawn in Figure 17.

Figure 16. Eutectic-like model of crystallization within a stack of crystalline lamellae, showing the diffusion paths of A (crystallizable) and B (uncrystallizable) molecules. Here L is the long period and Lc is the crystal thickness.

crystals, with the growth direction vertical. The thickness of the lamellar crystals is fixed uniquely by the undercooling. In order for a crystallization front to propagate, chains of the crystallizable component A must migrate to the growing tips of A-lamellae, while chains of the uncrystallizable component B must migrate away from the A-tips. For a sufficiently low diffusion length, the migration of B will be largely lateral, into the interlamellar regions. If the interlamellar amorphous layers are very thin, the compositional fields of neighboring crystal lamellae will overlap, and ultimately the system will behave like a planar front, the concentration adjacent to the front building with time and producing ever slower parabolic kinetics. However, the width of the amorphous layer depends on how much A, as well as B, it contains. So the amorphous layer has a variable thickness, which should establish the nature of the concentration field and consequently the velocity of the crystallization front.

Figure 17. Computational cell for lamellar spacing simulation. A periodic structure, comprised of lamellae of thickness Lc, stacked periodically with a periodicity L, is represented by this unit cell. The unit cell contains a half period. Symmetric boundary conditions are applied at the lateral edges of the cell. Reproduced with permission from: Kit, K. M.; Schultz, J. M. Macromolecules 2002, 35, 9819.

This is a periodic structure, comprised of lamellae of thickness Lc, stacked periodically with a periodicity L. The simulation volume contains a half period. Symmetric boundary conditions are 6311


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high value of Lc/L is corresponds to the case in which there is no admixture of A in the intervening layer. For instance, for cA = 0.75, the value of Lc/L at high Peclet numbers is 0.75. In this case (higher undercoolings), the diffusion distance is too short to allow substantial enrichment of A from the far field into the intervening layer. The constant Lc/L at low Peclet numbers represents an optimal enrichment of A in the intervening layer when the diffusion distance has reached a sufficiently high value. The Peclet number is inverse to the diffusion length, δ = D/V, which decreases strongly with temperature. This means that there should be a relatively sharp transition (i.e., over about one Peclet number decade) from a situation in which, at low diffusion distance or high Pe, essentially all the crystallizable molecules are contained in the crystalline lamellae, to a low-Pe situation in which the intervening layer is greatly expanded through inclusion of crystallizable molecules. Certainly the magnitude of the interaction parameter will influence this behavior, but this has not yet been modeled. Figure 20 is a compilation of Lc/L data for several blends. The data have been treated in the following way. It is known that neat

applied at the lateral edges of the simulation volume. Lc/L can also be thought of as the linear crystallinity of the system. Again, the diffusion equation is solved everywhere beyond the crystals at each time step Δt, and the interface is stepped forward at the local velocity corresponding to the temperature, L and Lc, according to composition locally at the interface. One has thus generated the crystal growth velocity as a function of these three variables. Approximating the rate of mass transformation as the rate of volume transformation, the rate would be VLc/L. In this context, we seek the maximum in this quantity. Results of the simulation for an overall cA of 0.5 of the mass transformation rate versus Lc/L for several values of the Peclet number LV/D are given in Figure 18. In each case, a maximum mass transformation rate, at a specific value of Lc/L, is found, as expected.

Figure 18. Mass transformation rate versus the linear crystallinity (φc = Lc/L) of a 50/50 sPS/aPS blend, for several values pf the Peclet number. Reproduced with permission from: Kit, K. M.; Schultz, J. M. Macromolecules 2002, 35, 9819.

Figure 19 shows, for three values of overall composition, the Lc/L position corresponding to the maximum mass trans-

Figure 20. Normalized lamellar spacing vs overall concentration for several polymer blend systems. Note that all data fall on the lines representing the simulated solutions for large and small Peclet numbers. Reproduced with permission from: Kit, K. M.; Schultz, J. M. Macromolecules 2002, 35, 9819.

polymers exhibit an inherent Lc/L (possibly reflecting a response to stress fields). This inherent Lc/L is subtracted from the measured value, leaving only that portion of Lc/L which has been affected by the compositional field. This modified value is plotted versus composition. What is seen is that the modified Lc/L values fall onto either of two curves which represent the asymptotic high and low values, for high and low Peclet numbers. This result is consistent with the predicted Pe-dependent selection of wide or narrow intervening layers. F.3. Analytical Treatment: The Modified Zener Model. To date, analytical treatments are very limited. We have seen that the usual morphology for polymers crystallizing from the melt is that of stacks of crystals, the crystals separated from each other by noncrystalline matter. Morphologically and phenomenologically the situation for transformation in eutectoid and eutectic systems, reviewed in section E.4, is quite similar. In these cases, alternating platelets of phases α and β are formed from either a homogeneous liquid L (the eutectic case, L → α + β) or a homogeneous solid γ (the eutectoid case, γ → α + β).

Figure 19. Effect of Peclet number on the transformation rate maximum for the simulation of lamellae growing in a 50/50 sPS/aPS blend. Reproduced with permission from: Kit, K. M.; Schultz, J. M. Macromolecules 2002, 35, 9819.

formation rate plotted against Peclet number. We note specially that above and below the transitional Pe range from log Pe ≈ −1/2 to log Pe ≈ +1/2 (Pe between 0.3 and 3.2) the value of Lc/L is constant: a high level at high Pe and a low level at low Pe. The 6312


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Here, L and γ contain components A and B, and the new phases α and β are respectively richer and poorer in B than the parent L or γ. As the transformation front moves forward, B must move away from the propagating α front and A must move toward it. Similarly, A must move away from the β front and B must move toward it, as sketched in Figure 9. The distances over which these movements must occur is minimized if the α and β crystals are in the form of thin lamellae stacked alternately. The largest distance an A or B moiety need move then (laterally to the growth direction) is λ/2, where λ is the periodicity of stacking. Zener’s analytical model of this situation is based on the principle that A and B moieties can be rejected by the growing crystals only as fast as they can be transported away; the growth velocity and the transport of material in the original phase are coupled. This situation is similar to the crystallization from a binary polymer blend in which one or both components may crystallize. Here the components must also redistribute in the melt near the growth surface, and the most efficient means for this to take place is that in which there is an intimate stacking of growth arms of one phase, separated by either noncrystalline matter relatively rich in the uncrystallizable material or by growth arms of the other crystalline component. For these cases, one can set up a coupled growth-transport model, just as in the Zener model, incorporating later improvements in that model. The important distinction between the original eutectoid work and the polymer blend analogue is this: for the eutectoid or eutectic case, the growth velocity was free to take any value, whereas for polymers, which are always highly undercooled, the velocity is constrained by the kinetics of attachment, according to eq 4. In the polymer blend case, the velocity of the interface is written V ≅ V0(κ , T , c B∞)(1 − c Bint)

Figure 21. Computational cell used in the analytical model for eutecticlike crystallization of growth arms. Here regions 1 and 2 are the growth arm and the intervening amorphous region, respectively, f is the volume fraction of growth arms, and λ is the interarm spacing. Reproduced with permission from: Balijepalli, S.; Schultz, J. M. Macromolecules 1996, 29, 6601.

We seek a solution for the case of small Peclet numbers. The reason for this is as follows. The Peclet number is Pe =

where δ is the diffusion length. (We note that the Peclet number is inverse to the diffusion length.) Conceptually, λ is a measure of how far B molecules must diffuse to escape the advancing front, while δ is a measure of how far they are able tom diffuse. If the Peclet number is large, λ ≫ δ, and diffusion occurs only in a small region near the growing tip. This is the case for isolated arms and is of no interest. On the other hand, when λ ≤ δ, the compositional fields of neighboring arms overlap and the growth is strongly coupled. We explore a solution only the case for small Pe. This solution provides a value of cint B , allowing one to then fit the right and left sides of (4) and consequently giving the velocity V as a function of the periodicity λ. Figure 22 shows a

(4a)

where V0, the natural velocity is given by eq 2, using a Lauritzen− Hoffman type expression, and cint B is the local concentration of the solutal species at the interface at steady state. The cint B dependence on the right side of (4a) is, in addition to the (1 − cint ) B dilution term, in T†m, the melting point of the melt adjacent to the growth surface: Tm† = Tm* − mc Bint

(29)

* is the equilibrium where m is the slope of the liquidus curve and Tm melting point of the neat polymer, but adjusted for the capillarity of the interface. Balijepalli and Schultz have modeled the propagation of growth arms into a melt,107 generally following the analysis of Zener. The computational cell used in the analytical model is sketched in Figure 21. Here regions 1 and 2 are the growth arm and the intervening amorphous region, respectively, f is the volume fraction of growth arms, and λ is the interarm spacing. The growth arms and the intervening layers are assumed to be extensive in the x-direction. Growth occurs along z. Knowing that the spherulite will grow at steady state, one can write the diffusion equation as ⎛ ∂ 2c ∂ 2c B ⎞ ∂c ⎟=V B −Dm⎜ 2B + 2 ∂z ∂z ⎠ ⎝ ∂y

λ Vλ = δ Dm

Figure 22. Computed interface velocity versus growth arm spacing for a 70/30 high/low molecular weight PEO blend crystallizing at 54 °C. Reproduced with permission from ref 107.

(30)

In (30), the left side represents the flux of component B away from the growing A-arms, while the right side represents the rate of exclusion of B from the growing A-arms. The situation modeled here differs from the original Zener model mainly in that the growth velocity is specified by conditions at the growth surface (eq 4) and is no longer freely variable.

representative result. Seen are the analytical solution, representing a 70/30 blend of a high and a low molecular weight poly(ethylene oxide) (PEO) and also a point representing the measured growth velocity and arm spacing λ. The measured value is very near the maximum in the analytical result. This finding allows one to postulate that the morphology is selected 6313


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such as to provide the maximum growth velocity (a not unreasonable concept). It should be mentioned that the initial steep increase of velocity with arm spacing is a capillarity effect. At very small arm spacing λ, the positive interfacial energy at the front is competitive with the free energy decrement due to the phase change, thereby severely decreasing the undercooling. This effect vanishes linearly with arm spacing at small values of λ and is finally overwhelmed by the effect of overlapping fields at higher λ. Figure 23, for the same 70/30 blend and also for an 80/20 blend, shows the predicted growth velocity versus crystallization

Figure 24. Calculated and experimental growth arm spacings versus crystallization temperature for an 80/20 high/low molecular weight PEO blend. Reproduced with permission from ref 107.

parameter ψ is also taken to vary continuously from a crystalline phase to a noncrystalline phase, the order parameter dropping from a value of 1 in a (perfect) crystal to 0 in a noncrystalline phase over some (normally very short) distance (the “interface thickness”). The order parameter can be related to the local entropy, and in this way use can be made of equilibrium and irreversible thermodynamics. Phase-field modeling has been applied to two-dimensional spherulitic polymer crystallization (as for polymer on a on a microscope slide).112−114 The specific application of the phase-field method varies in detail from investigator to investigator, but all follow the same general route. In the description below, we follow Mehta and Kyu.126 The method begins by expressing the energy density of the transforming system as a function of the crystal order parameter ψ. This is always expressed as a double-well system, as shown in Figure 25, such that below the melting point T0m the energy density at ψ = 1 (the perfect crystal) is lower than the minimum at ψ = 0 (the melt or solution) and conversely above the melting point. A mathematically convenient expression for the point by point free energy of local order is ⎡ ψ2 ψ3 ψ4 ⎤ flocal (ψ ) = W ⎢ζ − (1 − ζ ) + ⎥ 3 4 ⎦ ⎣ 2

(31)

Figure 23. Calculated and experimental values for spherulite growth velocities versus crystallization temperature for 80/20 and 70/30 high/ low molecular weight PEO blends. Reproduced with permission from ref 107.

As required, this expression (the Landau expansion) has minima at ψ = 0 (the melt) and at ψ < 1 (a partially ordered state) and also expresses a maximum at ψ = ζ. One expects also to have an energy contribution from the order gradient. The gradient term is usually expressed as

temperature, using the criterion of maximum growth velocity. Shown also are the measured values for several temperatures. The agreement of model and measurement is good at higher temperatures, where the Peclet number is low but deviates at lower temperatures (where the model loses accuracy). The growth arm spacing versus crystallization is shown in Figure 24. Here, at higher temperatures, the model and measurement differ by some 50% acceptable for this modelbut deviate considerably at lower crystallization temperatures, as did the growth velocity. F.4. Phase-Field Simulation. Phase-field modeling108−111 provides a methodology for exploring and understanding how morphologies evolve during phase transitions. In this approach, equations representing the local evolution of order (or crystallinity) and the local transfer of heat or chemical species are written and then coupled together. Like the transfer of heat and the diffusion of chemical species, the change in an order

[κ(ψ ) ·∇ψ ]2 (32) 2 In the phase field approach, as it is applied to polymer crystallization, the local free energy density f local(ψ) is coupled to another energy or entropy term. In the case of a polymer blend, one thinks immediately of concentration effects. Consider a two-component blend, with one component (A) crystallizable and the other (B) not. Internal energy considerations preclude the presence of an appreciable concentration of B in crystals of A. Consequently, the effect of B is on the free energy of the melt or solution. Using ϕ(r,t) to denote the local concentration of B, one writes the local free energy of the melt115 as fgrad (ψ ) =

floc (ϕ) =

1−ϕ ϕ ln(1 − ϕ) + ln(ϕ) + χϕ(1 − ϕ) nA nB (33)

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Figure 25. Variation of the local order free energy density as a function of crystal order parameter for several temperatures. The crystal state ψ = ζ0 varies with crystallization temperature. Reproduced with permission from ref 112. Copyright 2004 John Wiley and Sons, Inc.

Figure 26. Phase field simulations of the growth of a spherulite. Upper row: composition maps. A grayscale map was used to increase the contrast. Lower row: orientation map. Reproduced with permission from ref 114. Copyright 2005 American Physical Society.

where nA and nB are the numbers of statistical segments per mole of A and B, respectively, and χ is the Flory−Huggins interaction parameter. In addition to the local concentration effect, one expects also a gradient term, taken normally in the same form as (32): [κ(ϕ) ·∇ϕ]2 fgrad (ϕ) = 2

With the free energies now expressed, one can utilize Onsager relationships to write for the time dependences (at all points in space):116

∂ψ δF = −Γ ψ ∂t δψ and

Additionally, one may add a term specifying an effect of order and composition on each other. As an example, Mehta and Kyu use f ψϕ = −αψ (ϕ − ϕ0)

∂ϕ δF = ∇·Γ ϕ·∇ ∂t δϕ

(35)

⎛

2

∫ ⎜⎝floc (ψ ) + floc (ϕ) + f ψϕ + κ ψ |∇2ψ | + κϕ

|∇ϕ|2 ⎞ ⎟ dΩ 2 ⎠

(38)

Here Γψ and Δ·Γϕ are mobilities for crystallization and molecular transport, respectively. Equations 37 and 38 are now used to find simultaneously the concentration field and the position of the growth front at all times. This simulation allows one to follow the development of a spherulite from inception to any subsequent time. Figure 26 shows the simulated development of a polymer spherulite growing into the melt of a blend. The grayscale intensity in the top sequence depicts the compositional field. The color in the lower sequence shows the orientational field.

where φ0 is the initial concentration of the blend. The total free energy of the system can then be expressed as F(ψ , ϕ) =

(37)

(34)

(36)

where the integral is over all space. 6315


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to the mobility) has an algebraically more complex temperature dependence (eq 4), in which the driving force term, rather than the molecular mobility term, dominates. Further, the solute concentration is important in the Γψ term, via both a dilution factor and through the effects of concentration on the equilibrium melting point and the molecular mobility. There is, however, nothing to prevent these features from being built into the phase-field modeling of the crystallization of polymers from the melt. The direct finite element method, conversely, builds these effects in very well but has so far been rigid in forcing the geometries to conform to a predetermined pattern.

An analogous procedure can be used to simulate the effect of the release of the heat of fusion on spherulite development.127 Figure 27 shows the simulation of the growth of a pure

G. PATTERN FORMATION DURING FIBER SPINNING: A ROLE OF THERMAL FIELDS High-speed fiber spinning and the heat-setting process in conventional spun fibers are archetypal cases of crystallization under very high molecular orientation. In these cases, the crystal/ noncrystal growth front moves at velocities of at least 75 cm/s and up to some 15 000 cm/s, and crystallization is observed to occur at temperatures far below what is possible in quiescent crystallization. There are three very interesting implications here: 1 With “solute” diffusivities in the range of 10−14−10−10 cm2/s, the solute diffusion length at these growth rates becomes of atomic dimensions. This means that it is impossible for the noncrystallizable molecules to diffuse away fast enough to keep up with the growth front. Consequently, any normally noncrystallizable molecules, or sequences within a chain, must be incorporated into the growing crystals. The crystals so-produced must be highly defective and only metastable. It would require a high driving force to permit such massive defect incorporation. The situation is sketched in Figure 28. Here, defective

Figure 27. Phase field simulation of the growth of a pure sPP spherulite crystallized at 100 °C Shown here is the crystal order-parameter field. Reproduced with permission from ref 113. Copyright 2005 American Institute of Physics.

syndiotactic polystyrene spherulite growing at 100 °C. Here the growth arms form in response to the thermal field. While it is physically unlikely that thermal diffusion lengths could be small enough to produce the effects shown, the figure shows interesting detail regarding what the sequence of events could be under given conditions. As these figures show, phase-field simulations can very nicely depict possible paths of development of spherulites. One of the great advantages of the method is that shapes develop without hindrance. One sees in the above figures the development of the densely branched morphology suggested by Goldenfeld.78,79 Indeed, the potential of physically rigorous prediction of structures and structure development is very good. To date, however, simplifications in the simulations have precluded the use in quantitative prediction. The principal difficulty lies in how the interface mobility has been coupled to the solute or thermal field. In all cases to date, the order mobility term, Γψ, in (37) has been taken as inversely proportional to the viscosity. We know, however, that the natural interface velocity (which relates directly

Figure 28. Sketch of the growth of narrow, defective crystal fibrils in a hyperstretched, undercooled melt, as in high-speed spinning from the melt. Reproduced with permission from ref 18. Copyright 1984 John Wiley & Sons, Inc.

fibrillar crystals form as the heat of fusion is released and is dissipated, largely laterally, at the crystallization front. 2 The natural growth rate, requiring segmental transfer, is orders of magnitude too low to produce the necessary growth-front propagation rates. However, if a diffusionless 6316


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transformation is possible, crystallization can still occur. For such a diffusionless process to occur, at least the following must obtain: a The chain repeat units must be already near their ultimate crystalline sites, so that no long-range transport is necessary. This implies that the system be highly oriented; diffusionless processes should occur only in highly elongated systems. b Normally unincorporable components and chain imperfections must be included in the transformed product, unless their mobility is enormously greater than that of the normally crystallizing polymer. Because of the incorporation of such “defects”, the product will be metastable and should be capable of further transformation later. Since the motion of the crystallizing molecules is so restricted, kinks and jogs in the chain must be incorporated. Experimentally, it is found that the crystals initially formed exhibit poor crystalline order.117−120 With heat treatment, the defects rapidly migrate, causing formation of a periodic crystal−amorphous sequencing. Figure 29 shows dark-field transmission

Figure 30. SAXS curves taken during the annealing of an isotactic polypropylene film during annealing at 90 °C. Reproduced with permission from ref 123. Copyright 1982 John Wiley & Sons, Inc.

regions axially along the fibril.123 The suggested process is illustrated in Figure 31. The defects used

Figure 29. Isotactic polystyrene films prepared by drawing from the melt according to the Gohil−Petermann method: left, as drawn; right, after annealing near the melting point.

Figure 31. Stages in the annealing of a defective fibrillar crystal: left, depiction of the axial migration of defects; right, depiction of the axial density gradient during the process. Reproduced with permission from ref 123. Copyright 1982 John Wiley & Sons, Inc.

electron micrographs of a melt-drawn isotactic polystyrene film in the as-drawn state and after annealing near the melting point. The long, fibrillar crystals of poor contrastindicating defective crystalsin the as-drawn state transform to rows of small, more perfect crystals after annealing. Small-angle X-ray scattering (SAXS) results of such transformations have proved useful. Figure 30 shows a nest of SAXS curves from a melt-drawn isotactic polypropylene film annealed at 90 °C for various times. An interesting feature of this nest of curves is that the curves intersect and invert at a point (marked by an arrow) above the position of the maximum intensity. This is a hallmark of spinodal transformation,121,122 strongly suggesting that the transformation occurs by a gradual, spinodal-like process in which intrachain defects migrate to form small defect-free and defect-rich

in this illustration are chain ends and chain kinks. Initially, the defects would be distributed randomly in the imperfect fibrillar crystal. Upon heating, the defects migrate axially, producing a sinusoidally varying density distribution whose amplitude increases with time. A similar result has been reported by Cakmak et al.,124 and a simulation by Meakin and Scalapino has confirmed the plausibility of this process.125 c The effective undercooling must be very great, at least locally, in order to produce a net free energy decrease upon crystallization to the metastable product. The large driving force is largely due to the very high molecular orientation, which increases the melting point. The additional undercooling due to orientation is tens or hundreds of degrees.18,126 6317


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d A molecular mechanism of diffusionless transformation must be available. It is suggested that in these cases the chains need only “jostle” slightly to come into near-crystalline registry. A more precisely defined mechanism has not been proposed. 3 The transformation kinetics must become governed by the rate at which the heat of fusion can be removed. Suppose that crystallization is taking place at a velocity of 75 cm/s, as directly observed in postspinning treatment of PET fibers.127 With a thermal diffusivity of 6 × 10−4 cm2/s, the thermal diffusion length will be somewhat less than 10 nm. For PET fibers spun at 10 000 cm/s (in the ultrahighspeed spinning range), the thermal diffusion length will be under 1 nm. Thus, it is the thermal field which should be operative here. Given the small magnitudes of the thermal diffusion lengths, and since crystal growth-front dimensions are generally of the order of magnitude of the diffusion length, very fine filamentary crystals are expected. 4 Finally, for molecular orientation to be effective, crystallization must occur before relaxation to a less oriented state can occur. For PET, such relaxation is found to occur in a time of the order of 10−100 ms.131,128 Consequently, processes in which such diffusionless transformations occur must be such that the molecules are in a sufficiently extended state during the crystallization process. Normally this requires a positive stress or strain on the system during the transformation. Analytically, the problem has been structured as a variant on Ivantsov’s solution (eq 25) for rodlike crystallization with the liberation of the heat of fusion For long-chain polymers, an undercooling term, δT, representing the effect of chain extension on the melting point, must be incorporated in the right-hand side of (25):

Figure 32. Computed spinline velocity versus crystal fibril radius for PET spun at a range of temperatures. Reproduced with permission from ref 129. Copyright 1991 John Wiley & Sons, Inc.

−ye y E i( −y) = (cpTs/L)/[1 − (T0 + δTs)/Tm − (2γs/L)/ρ − δT (V )/Tm]

Figure 33. Comparison of predicted and measured crystalline fibril diameters versus spinline velocity for PET spun from the melt. Reproduced with permission from ref 129. Copyright 1991 John Wiley & Sons, Inc.

(39)

Algebraically, the effect of chain extension enters as a simple (but very large) decrement to the far-field temperature T0. In the case of fiber spinning, the molten, highly extended fiber is effectively moved through a temperature gradient (constrained growth) at a given velocity. If the final product is crystalline, crystal growth had to have taken place at some point along the line such that V is exactly equal to the local threadline velocity. Thus, this steady-state V has its value imposed externally, and (39) can be solved directly to obtain a ρ−T relationship for any threadline velocity (ρ being the tip radius of the growing fibril). An approximate solution to (39), using parameters appropriate to poly(ethylene terephthalate) (PET) spinlines and using an optimization principle akin to minimum undercooling, has yielded the results shown in Figure 32.129 Consider a specific spinline velocity, say 4500 m/min, and follow what happens as a volume element of the molten fiber moves to lower and lower temperatures. At 210, 200, 190, and 185 °C, no compatible solution is possible. At 184 °C, crystallization has become possible, at an initial fibril radius of some 12 nm. Taking this as the operating condition, one can find the temperature of crystallization and the crystal fibril radius at any spinline velocity. Figure 33 shows the computed crystallization temperature and crystal tip radius as functions of the spinline velocity. Careful X-ray radial distribution results on spun PET fibers confirm the nanometer radii of the crystal fibrils.133 The predicted fibril radius

and crystallization temperature are of the correct magnitudes and vary in the right direction with spinline velocity; i.e., reasonable agreement with the experimental results of Heuvel and Huisman,130 Perez,131 Shimizu et al.,132,133 and George134 is found.

H. STRESS FIELDS IN QUIESCENT MELTS: CONJECTURES ON THEIR ROLE There has been little investigation directed specifically at the effects of stress fields (in the adjacent melt) on crystallization from polymer melts. We shall see first that there is good reason to believe that such effects exist and that they are important. We will then review a recent set of investigations which probe stress field effects through the molecular mass dependence of spherulite details. These results show that stress fields are very important. There are several aspects of spherulitic crystallization that hint at stress field effects. Given below are several of those indications. 1 We have seen above that the morphology internal to spherulites is dictated in crystallizable blends by compositional fields. But spherulitic crystallization occurs also in chemically homogeneous, monodisperse polymers. If a field determines spherulite formation in blends, and if the 6318


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same generic morphology occurs in “pure” polymers, one should first seek explanation through some kind of field, and common to all polymers above the entanglement limit are stress fields. 2 Spherulites are observed in inorganic and organic materials, as well as in polymers, but in all cases the melts or solutions are highly viscous. Stress fields can exist in liquids only if the fluid is highly viscous. This again is suggestive of a role of stress fields. 3 Where the stress field is enhanced, the spherulite growth velocity decreases steeply. For example, the stress field is enhanced in the melt as three spherulites grow together in a molten polymer film. In this occluded volume, the stress fields of the three spherulites overlap and reinforce each other. As the remaining volume of melt becomes very small, the stress is great enough to create profuse cavitation, which can be seen in the optical microscope135 and heard as acoustic emission signals.136,137 Most significantly, measurements on isotactic polypropylene (iPP) and on poly(ethylene adipate) (PEA) show that prior to cavitation the growth velocity decreases continuously, by a significant factor (ca. 1/9 for iPP crystallizing at 127 °C and 1/3 for PEA crystallizing at 31 °C).138 Following cavitation, the growth velocity returns to nearly its original, higher level.138 These effects are seen in Figure 34.

Figure 35. Amorphous layer thickness versus the square root of the molecular weight for polyethylene rapidly crystallized from the melt. Reproduced with permission from ref 139.

Figure 34. Effect of self-generated stresses on spherulite growth kinetics. (a) growth velocity of isotactic polypropylene spherulites in the region remaining as three spherulites grow together at 127 °C. Here relative volume refers to the remaining volume of melt as the spherulites grow toward each other. In (a) the crystallizing film is uncovered and no cavitation occurs; in (b), the film is covered and cavitation occurs at a relative volume of ca. 0.25. Reproduced with permission from ref 138. Copyright 2003 John Wiley & Sons, Inc.

other words, R-S-R associate the thickness of the amorphous layer with an equilibrium between the driving force for crystallization and the creation of stress in the amorphous layer. Since most morphogical features depend on the kinetics of crystallization, it is likely that the balance is really a kinetic one: the rate of free energy decrease due to crystallization balances the rate of increase of strain energy, just as we have seen for composition and heat, and dictates the thickness of the amorphous layer. There is, however, no extant model for such a process. Toda and co-workers, in a recent series of publications,141−143 have produced evidence of the role of self-generated stress fields in spherulite formation. Their work makes use of a finding by Briber and Khoury,144 suggested also in the work of others,145,146 that the twisting and branching of ribbonlike lamellae during spherulite growth from the melt are coupled; twisting is localized near branches, and the period of the twist decreases with decreasing frequency of branching. Toda and co-workers begin by noting that a Mullins and Sekerka type of surface instability should occur during the growth of the ribbonlike crystals forming a spherulite, when some selfgenerated field reaches a critical value. This instability would occur at a specific lamellar width λ. One next assumes (as seems reasonable) that the crystal grows laterally at some constant rate proportional to the axial growth velocity. Then when the width reaches the value λ, the growing ribbon bifurcates, as sketched in Figure 36. As these bifurcated crystals continue to grow axially

4 There appears to be an effect at the interlamellar level. Robelin-Souffaché and Rault find a significant increase with molecular weight of the thickness la of the amorphous layer separating lamellar crystals within spherulites of polyethylene formed at the same undercooling,139 as shown in Figure 35. Magill et al. have shown, using poly(tetramethyl-p-silphenylene siloxane), that the effect of this molecular weight effect on la persists over a wide range of crystallization temperature.140 Both sets of authors have attributed this effect to stresses associated with chain entanglements trapped between adjacent lamellae. Robelin-Souffaché and Rault (R-S-R) write, “The balance between the enthalpy of crystallization and the energy of deformation of the amorphous layers explains the correlation law L ∼ la ∼ r [L is the long period and r the end-to-end distance of the chain], the variation of L with supercooling, and the order of magnitude of the crystallinity for high molecular weight materials ....” In

Figure 36. Schematic illustrations of (a) the evolution of crystal branching at the critical width, λ, and the reorientation on splaying and (b) the expected structure. Reproduced with permission from ref 142. 6319


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⎛ x v γ D ⎞1/2 λc = 2π ⎜ 1 s ⎟ ⎝ kBΔc VT ⎠

and laterally, they must overlap each other, as also sketched in Figure 36. Upon overlapping, giant screw dislocations are formed, a process documented in polymer crystallization over 50 years ago.2 Associated with each screw dislocation is an element of twist φ.147 The process of ribbon bifurcation and dislocation formation must occur periodically as the ribbonlike crystal grows. If the dislocations all have the same hand (an autocatylitic process resulting from the dislocation’s strain field within the crystals), then after 2π/φ repetitions, a full 2π twist will have been produced. This behavior is sketched in Figure 37.

and λs = 2πb

1/2 vs ⎛ γ 1 ⎞ ⎜ ⎟ Δv ⎝ 12 Vη ⎠

(42)

where x1 is the number of segments of the fractions rejected at the growth front, Δcs the difference between the actual concentration at the growth front and the far-field value, D the diffusivity, V the growth velocity, b the shear flow thickness, Δv the specific volume difference between crystal and melt, and η the shear viscosity. Since the lateral growth velocity is proportional to the axial growth velocity, xc is proportional to λ. Hence, the periodicity P of the twist must be proportional to λ. Thus P∝λ∝

⎛D⎞ ⎛ 1 ⎞ ⎜ ⎟ or ⎜ ⎟ ⎝ V ⎠ ⎝ Vη ⎠

(43)

for compositional or stress fields, respectively. From Einstein’s equation connecting diffusivity and viscosity, the right-hand sides of eq 43 have the same temperature dependence. However, the molecular mass dependences are quite different: for mass diffusion and for viscosity148

Figure 37. Sketch of a regular sequence of giant screw dislocations in a crystalline polymer ribbon. Such an array causes a a twist about the ribbon axis. Reproduced with permission from ref 147. Copyright 1969 Elsevier

D ∝ M −2 and

The period P of the twist is thus 2πxc/φ, where xc is the spacing between the giant screw dislocations. The surface instability could be generated by either a composition field or a stress field. In either case, the instability will occur when the crystal width reaches the critical level ⎛ v γ ⎞1/2 λ = 2π ⎜ s ⎟ ⎝a ⎠

(41)

1 ∝ M −3.6 η

(44)

Thus, testing the molecular weight dependence of the periodicity should be useful in determining which kind of field is dominant in lamellar branching and twisting. Further, for a polydisperse homopolymer, the only composition field is that from the polydispersity of molecular weight. The dependence of twist period on polydispersity can then be a qualitative test of the mechanism. Finally, blending a crystalline polymer with a chemically identical uncrystallizable polymer can be useful in differentiating the effects of different fields, as we shall see. The results of the experiments of Toda et al. indicate strongly that that the stress field controls the branching and twisting of ribbonlike lamellae during crystallization from the melt. In quantitative studies of the molecular weight dependence of twist periodicity, the periodicity P times the square of velocity V is

(40)

where vs is the specific volume of a segment in the crystal, γ is the crystal-melt surface energy, and a is the gradient of composition or stress in the melt adjacent to the growth surface. Using expressions for compositional and stress gradients, Toda et al. find the following for the critical widths λc and λs for compositional and stress fields, respectively:

Figure 38. Double-logarithmic plots of the product, P2V, against Mw. (a) Results for several molecular weights and with polydispersities ranging from 1.1 to 3.3, measured at the fixed crystallization temperature of 116 °C. The slope of the fitting line is −2.7. (b) Results for 1:1 crystallizable/noncrystallizable blends, in which the crystallizable polyethylene has molecular weight 32 100 and the molecular mass of the uncrystallizable hydrogenated butadiene is varied. A plot of λm2V (▲) against Mw is also shown. The crystallization temperature was fixed at 116 °C. The slopes of the fitting lines are −3.9 and −3.6 for P2V and λm2V, respectively. Reproduced with permission from ref 143. 6320


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Biography

plotted versus molecular weight M on a double-logarithmic scale. Equations 43 and 44 tell us that data should lie on straight lines and that the slopes of these lines should be −2 or −3.6. Toda et al.’s first results142,143 for neat poly(vinylidene fluoride) and polyethylene of various molecular weights and polydispersities were inconclusive. The data for a number of molecular weights and weight distributions (ranging from Mw/Mn = 1.1 to 3.3) are shown in Figure 38a.143 The data fell very nicely on straight lines, as expected. The insensitivity of the result to molecular weight distribution is a hint that compositional fields are not controlling. However, the slopes of those lines were nearer to −2 than to −3.6. It is possible, however, that in changing the molecular weights of these homopolymers, more than changes in only the diffusivity and viscosity of the melt were altered. It is likely that changes in details of crystalllization, such as the morphology of the lamellae, the amount of twist per dislocation, and the ratio of transverse and longitudinal growth velocities, were altered. To better test the situation, a sharp fraction of polyethylene (Mw/Mn < 1.1) was blended with uncrystallizable hydrogenated polybutadienes of various molecular masses, in order to always use the same crystallizing polymer, but varying the diffusivity and viscosity of the melt.3 Figure 38b shows this result. The slope of the curve is −3.9, close to what is expected for stress field control. These results provide the most direct evidence to date regarding the influence of stress fields in the melt on polymer crystallization

Jerold M. Schultz is C. Ernest Birchenall Professor, Emeritus in the Department of Chemical and Biomolecular Engineering and the Department of Materials Science and Engineering at the University of Delaware. He received his Ph.D. in Metallurgical Engineering from Carnegie Mellon University and holds B.S. and M.S. degrees in the same field from the University of California. Prof. Schultz is a fellow of the American Physical Society. He has been until recently an associate editor of Macromolecules. Prof. Schultz has held named visiting professorships at the University of Bristol and at Akron University and is honorary professor at the Beijing University of Chemical Technology. He was awarded the Kliment Ohridski medal in Bulgaria for his work. He is the author of textbooks on polymer materials science and on diffraction theory and of a monograph on polymer crystallization as well as having jointly edited several books in the field of polymers. He has published over 200 papers in scientific journals. Prof. Schultz’s interests have been in the structure and properties of polymeric solids and of how these relate to physical processing. He has also maintained interests more generally in diffraction and scattering and in the mechanical behavior of materials.

I. SUMMARY General concepts of the effects of compositional, thermal, and stress fields have been reviewed. We have seen that models which have been developed for small molecule systems are well developed and satisfactorily predict morphological detail. The solutions to moving boundary problems have produced quantitatively predictive results for small-molecule systems and have shown the utility of the diffusion length and the Peclet number for crystallization. Models for the crystallization of eutectic and eutectoid systems have shown that the coupled growth of alternately stacked thin crystalline plates is necessary to maximize the rate of free energy decrease. These concepts and models have been adapted to explain and predict the development of morphological detail in crystallizing polymer systems. Compositional fields have been shown to quantitatively determine the growth arm thickness and spacing in the crystallization of polymer blends. Thermal fields have been shown to dictate the internal structure of fibers spun from the melt at high speed or crystallized as an after-treatment to spinning. It is strongly indicated that stress fields in the melt play an important role in defining intraspherulite morphology. Phase-field simulations have been adapted from the metals field to show how morphology develops in the spherulitic crystallization of polymers. Most of the extant simulations (finite element, analytical, and phase-field modeling) have been performed at relatively rudimentary levels; greater sophistication will be needed to render these approaches satisfactory for engineering application. An important result of self-generated fields (when they are operative) is that morphological detail at the appropriate length range scales proportionately to the diffusion length.

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REFERENCES

(1) Libbrecht, K. G. Rep. Prog. Phys. 2005, 68, 855. (2) Reneker, D. H.; Geil, P. H. J. Appl. Phys. 1960, 31, 1916. (3) Geil, P. H.; Reneker, D. H. J. Polym. Sci. 1961, 51, 569. (4) Langer, J. S. Rev. Mod. Phys. 1980, 52, 1. (5) Laxmanan, V. Acta Metall. 1985, 13, 1023. (6) Schultz, J. M. Polymer Materials Science; Prentice-Hall: New York, 1974. (7) Martuscelli, E.; Canetti, M.; Vicini, L.; Seves, A. Polymer 1982, 23, 331. (8) Silvestre, C.; Cimmino, S.; Martuscelli, E.; Karasz, F. E.; MacKnight, W. J. Polymer 1987, 28, 1190. (9) Keith, H. D.; Padden, F. J., Jr. J. Appl. Phys. 1963, 34, 2409. (10) Keith, H. D.; Padden, F. J., Jr. J. Appl. Phys. 1964, 35, 1270. (11) Keith, H. D.; Padden, F. J., Jr. J. Appl. Phys. 1964, 35, 1286. (12) Keith, H. D.; Padden, F. J., Jr. J. Polym. Sci., Polym. Phys. Ed. 1987, 25, 229. (13) Keith, H. D.; Padden, F. J., Jr. J. Polym. Sci., Polym. Phys. Ed. 1987, 25, 2265. (14) Calvert, P. D. J. Polym. Sci., Polym. Phys. Ed. 1983, 21, 467. (15) Goldenfeld, N. J. Cryst. Growth 1987, 84, 601. (16) Tanaka, H.; Nishi, T. Phys. Rev. Lett. 1985, 55, 1102. (17) Schultz, J. M. Polymer 2003, 44, 433. (18) Tiller, W. A.; Schultz, J. M. J. Polym. Sci., Polym. Phys. Ed. 1984, 22, 143. (19) Murray, R.; Davis, H. A.; Tucker, P. J. Appl. Polym. Sci. 1978, 33, 177.

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest. 6321


Macromolecules

Perspective

(68) Somboonsuk, K.; Trivedi, R. Acta Metall. 1985, 33, 1051. (69) Esaka, H.; Kurz, W. J. Cryst. Growth 1985, 72, 578. (70) Oswald, P.; Bechhoefer, J.; Libchaber, A. Phys. Rev. Lett. 1987, 58, 2318. (71) Ben-Jacob, E.; Goldenfeld, N.; Langer, J. S.; Schön, G. Phys. Rev. Lett. 1983, 51, 1930. (72) Ben-Jacob, E.; Goldenfeld, N.; Langer, J. S.; Schön. Phys. Rev. A 1984, 29, 330. (73) Ben-Jacob, E.; Goldenfeld, N.; Kotliar, B. G.; Langer, J. S. Phys. Rev. Lett. 1984, 53, 2110. (74) Brower, R. C.; Kessler, D. A.; Koplik, J.; Levine, H. Phys. Rev. A 1984, 29. (75) Hartin, O.; Goldenfeld, N. Phys. Rev. A 1987, A35, 1382. (76) Kessler, D. A.; Koplic, J.; Levine, H. Phys. Rev. A 1984, 30, 3161. (77) Kessler, D. A.; Koplic, J.; Levine, H. Phys. Rev. A 1986, 33, 3352. (78) Goldenfeld, N. In Metastability and Incompletely Posed Problems; Antman, S., Ericksen, J. L., Kinderlehrer, D., Müller, I., Eds.; Springer: New York, 1987. (79) Goldenfeld, N. In Physicochemical Hydrodynamics; Velarde, M. G., Ed.; Plenum: New York, 1988. (80) Liu, F.; Goldenfeld, N. Phys. Rev. A 1988, A38, 407. (81) Glicksman, M. E.; Schaeffer, R. J.; Ayers, J. D. Metall. Trans. 1976, A7, 1747. (82) Langer, J. S.; R. F. Sekerka, R. F.; Fujioka, T. J. Cryst. Growth 1978, 44, 414. (83) Huang, S. C.; Glicksman, M. E. Acta Metall. 1981, 29, 701. (84) Huang, S. C.; Glicksman, M. E. Acta Metall. 1981, 29, 717. (85) Ben-Jacob, E.; Deutscher, G.; Garik, P.; Goldenfeld, N.; Lareah, Y. Phys. Rev. Lett. 1986, 57, 1903. (86) Kallungal, J. P.; Barduhn, A. J. AIChE J. 1987, 36, 294. (87) Trivedi, R. Acta Metall. 1984, Al5, 977. (88) Oswald, P.; Bechhoefer, J.; Libchaber, A.; Lequeux, F. Phys. Rev. A 1987, 36, 5832. (89) Bechhoefer, J.; Oswald, P.; Libchaber, A.; Germain, C. Phys. Rev. A 1988, 37, 1691. (90) Laxmanan, V. J. Cryst. Growth 1987, 83, 391. (91) Zener, C. Trans. AIME 1946, 167, 550. (92) Hillert, M. Jernkontorets Ann. 1957, 141, 757. (93) Jackson, K. A.; Hunt, J. D. Trans. AIME 1966, 236, 1129. (94) D. Turnbull, D. Acta Metall. 1955, 3, 55. (95) Magnin, P.; Kurz, W. Acta Metall. 1987, 35, 1119. (96) Trivedi, R.; Magnin, P. Acta Metall. Mater. 1991, 39, 453. (97) Kurz, W.; Fischer, D. J. Fundamentals of Solidification; Trans Tech Publications: The Netherlands, 1992. (98) Magnin, P.; Mason, J. T.; Trivedi, R. Acta Metall. Mater. 1991, 39, 469. (99) Jincheng, L.; Elliott, R. Acta Metall. Mater. 1995, 43, 3301. (100) Tiller, W. A. The Science of Crystallization: Macroscopic Phenomena and Defect Generation; Cambridge University Press: Cambridge, UK, 1991; pp 308−314. (101) Burden, M. H.; Hunt., J. D. J. Cryst. Growth 1974, 22, 109. (102) Allen, D. J.; Hunt, J. D. Metall. Mater. Trans. A 1979, 10, 1389. (103) Kurz, W.; Fisher, D. J. Acta Metall. 1981, 29, 11. (104) Kit, K. M.; Schultz, J. M. J. Polym. Sci., Polym. Phys. Ed. 1998, 36, 873. (105) Kit, K. M.; Schultz, J. M. Int. J. Numer. Methods Eng. 1997, 40, 2679−2692. (106) Kit, K. M. Polymer 1998, 39, 4969. (107) Balijepalli, S.; Schultz, J. M. Macromolecules 2006, 39, 7407. (108) Landau, L. D. Zh. Eksp. Teor. Fiz. 1937, 7, 19, 637. [English translation by ter Haar, D. Collected Papers of L. D. Landau; Gordon & Breach: New York, 1965; pp 193, 209]. (109) Chan, S.-K. J. Chem. Phys. 1977, 67, 5755. (110) Caginalp, G. In Applications of Field Theory to Statistical Mechanics; Garrido, L., Ed.; Springer: Berlin, 1985; p 216. (111) Langer, J. S. In Directions in Condensed Matter Physics; Grinstein, G., Mazenko, G., Eds.; World Scientific: Philadelphia, 1986; p 164. (112) Mehta, R.; Kyu, T. J. Polym. Sci., Polym. Phys. Ed. 2004, 42, 2892.

(20) H. Chang, H.; Lee, K.-G.; Schultz, J. M. J. Macromol. Sci., Phys. 1994, B33, 105. (21) Petermann, J.; Gohi1, R. M. J. Mater. Sci. 1979, 14, 2260. (22) Petermann, J.; Gohil, R. M. J. Polym. Sci., Polym. Lett. Ed. 1980, 18, 781. (23) Petermann, J. Makromol. Chem. 1981, 182, 613. (24) Petermann, J.; Gohil, R. M.; Schultz, J. M.; Hendricks, R. W.; Lin, J. S. J. Mater. Sci. 1981, 16, 265. (25) Rau, J.; Gohil, R. M.; Petermann, J.; Schultz, J. M. Colloid Polym. Sci. 1981, 259, 241. (26) Schultz, J. M.; Petermann, J. Colloid Polym. Sci. 1984, 262, 217. (27) Schultz, J. M.; Petermann, J. Colloid Polym. Sci. 1984, 262, 294. (28) Bassett, D. C. Principles of Polymer Morphology; Cambridge University Press: Cambridge, UK, 1981; Chapter 6. (29) Schultz, J. M. Polymer 1991, 32, 3268. (30) Lauritzen, J. I.; Hoffman, J. D. J. Res. Natl. Bur. Stand. (U. S.) 1960, 64A, 73. (31) Lauritzen, J. I.; Hoffman, J. D. J. Res. Natl. Bur. Stand. (U. S.) 1961, 65A, 297. (32) Prygocki, W. Int. Polym. Sci. Technol. 1974, 1, T/52. (33) Lauritzen, J. I. J. Appl. Phys. 1973, 44, 4353. (34) Hoffman, J. D.; Davis, G. T.; Lauritzen, J. I. In Treatise on Solid State Chemistry: Crystalline and Noncrystalline Solids; Hannay, N. B., Ed.; Plenum: New York, 1976; Vol. 3. (35) Hoffman, J. D.; Frolen, L. J.; Ross, G. S.; Lauritzen, J. I. J. Res. Natl. Bur. Stand. (U. S.) 1975, 79A, 671. (36) Kurz, W.; Fisher, D. J. Fundamentals of Solidification, 4th ed.; Enfield Publishing & Distribution Co.: Enfield, NH, 1998, (37) Stefan, J. Ann. Phys. 1891, 278, 269. (38) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids, 2nd ed.; Clarendon Press: Oxford, UK, 1986. (39) Wang, H.; Gan, Z.; Schultz, J. M.; Yan, S. Polymer 2008, 49, 2342. (40) Wang, T.; Li, H.; Wang, F.; Schultz, J. M.; Yan, S. Polym. Chem. 2011, 10, 1039. (41) Okada, T.; Saito, H.; Inoue, T. Polymer 1994, 35, 5699−5705. (42) Li, B.; Brody, H. D.; Kazimirov, A. Phys. Rev. E 2004, 70, 155. (43) Ikehara, T.; Kurihara, H.; Kataoka, T. J. Polym. Sci., Polym. Phys. Ed 2012, 50, 563. (44) Saratovkin, D. D. Dendritic Crystallization, 2nd ed. (translated from Russian by J. E. S. Bradley); Consultants Bureau: New York, 1959. References to Saratovkin’s earlier work, ca. 1950, are cited within. (45) Ivantsov, G. P. Growth of Crystals; Consultants Bureau: New York, 1958. (46) Horvay, G.; Cahn, J. W. Acta Metall. 1961, 9, 695. (47) Bolling, G. F.; Tiller, W. A. J. Appl. Phys. 1961, 32, 2587. (48) Nash, G. E.; Glicksman, M. E. Acta Metall. 1974, 22, 1283. (49) Dash, S. K.; Gill, W. N. Int. J. Heat Mass Transfer 1984, 27, l345. (50) Kind, M.; Gill, W. N.; Ananth, R. Chem. Eng. Commun. 1987, 55, 295. (51) Tirmizi, H.; Gill, W. N. J. Cryst. Growth 1987, 85, 488. (52) Temkin., D. E. Dokl. Akad. Nauk SSSR 1960, 132, 1307. (53) Mullins, W. W.; Sekerka, R. F. J. Appl. Phys. 1963, 34, 323. (54) Mullins, W. W.; Sekerka, R. F. J. Appl. Phys. 1964, 35, 444. (55) Sekerka, R. F. J. Appl. Phys. 1965, 36, 264. (56) Sekerka, R. F. In Crystal Growth; Peiser, H. S., Ed.; Pergamon: Oxford, 1967; p 733. (57) Sekerka, R. F. J. Phys. Chem. Solids 1967, 28, 983. (58) Caroli, B.; Roulet, B. J. Phys. 1983, 44, 945. (59) Caroli, B.; Caroli, C.; Roulet, B. J. Cryst. Growth 1984, 66, 575. (60) Caroli, B.; Caroli, C.; Roulet, B. J. Cryst. Growth 1984, 68, 677. (61) Caroli, B.; Caroli, C.; Roulet, B. J. Cryst. Growth 1985, 71, 235. (62) Caroli, B.; Caroli, C.; Roulet, B. J. Cryst. Growth 1986, 76, 31. (63) Oldfield, W. Mater. Sci. Eng. 1973, 11, 211. (64) Langer, J. S.; Müller-Krumbhaar, H. J. Cryst. Growth 1977, 42, 11. (65) Langer, J. S.; Müller-Krumbhaar, H. Acta Metall. 1978, 26, 1681, 1689, 1697. (66) Trivedi, R. J. Cryst. Growth 1980, 49, 219. (67) Somboonsuk, K.; Mason, J. T.; Trivedi, R. Metall. Trans. 1984, 15A, 967. 6322


Macromolecules

Perspective

(113) Xu, X.; Keawwattana, W.; Kyu, T. J. Chem. Phys. 2005, 123, 124908. (114) Gránásy, L.; Pusztai, T.; Tegze, G.; Warren, J. A.; Douglas, J. F. Phys. Rev. E 2005, 72, 011605. (115) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (116) Gunton, J. D.; San Miguel, M.; Sahni, P. S. In Phase Transitions; Academic Press: London, 1983; Vol. 8, p 267. (117) Gupte, K. M.; Motz, H.; Schultz, J. M. J. Polym. Sci., Polym. Phys. Ed. 1983, 21, 1927. (118) Peszkin, P. N.; Schultz, J. M.; Lin, J. S. J. Polym. Sci., Polym. Phys. Ed. 1986, 24, 2591. (119) Hristov, H. A.; Schultz, J. M. J. Polym. Sci., Polym. Phys. Ed. 1990, 28, 1647. (120) Hristov, H. A.; Barton, R.; Schultz, J. M. J. Polym. Sci., Polym. Phys. Ed. 1991, 29, 883. (121) Cahn, J. W. Trans. Metall. Soc. AIME 1968, 166, 242. (122) DeFontaine, D. In Ultrafine Grain Metals; Burke, J. H., Weiss, V., Eds.; Syracuse University: Syracuse, NY, 1970; p 93. (123) Petermann, J.; Gohil, R. M.; Schultz, J. M.; Hendricks, R. W.; Lin, J. S. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 523−534. (124) Cakmak, M.; Teitge, A.; Zachmann, H. G.; White, J. L. J. Polym. Sci., Polym. Phys. Ed. 1993, 31, 371. (125) Meakin, P.; Scalapino, D. J. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 179. (126) Ziabicki, A. In High Speed Spinning; Ziabicki, A., Kawai, H., Eds.; John Wiley: New York, 1985; p 225. (127) Hristov, H. A.; Schultz, J. M. Polymer 1988, 29, 1211. (128) Schultz, J. M. In Solid State Behavior of Linear Polyesters and Polycondensates; Schultz, J. M., Fakirov, S., Eds.; Prentice-Hall: Englewood Cliffs, NJ, 1990; p 75. (129) Schultz, J. M. Polym. Eng. Sci. 1991, 31, 661−666. (130) Heuvel, H. M.; Huisman, R. In High Speed Spinning; Ziabicki, A., Kawai, H., Eds.; John Wiley: New York, 1985; p 295. (131) Perez, G. In High Speed Spinning; Ziabicki, A., Kawai, H., Eds.; John Wiley: New York, 1985; p 333. (132) Shimizu, J.; Okui, N.; Kikutani, T. In High Speed Spinning; Ziabicki, A., Kawai, H., Eds., John Wiley: New York, 1985; p 429. (133) Shimizu, J.; Kikutani, T.; Takaku, A. In Proceedings of the International Symposium on Fiber Science and Technology, Hakone, Japan, 20−24 August 1985; Elsevier Applied Science: Barking, Essex, 1986; p 62. (134) George, H. H. Polym. Eng. Sci. 1982, 22, 292. (135) Magill, J. H. J. Appl. Phys. 1964, 35, 3249. (136) Galeski, A.; Koenczoel, L.; Piorkowska, E.; Baer, E. Nature 1987, 325, 40. (137) Galeski, A.; Piorkowska, E.; Koenczoel, L.; Baer, E. J. Polym. Sci., Polym. Phys. Ed. 1990, 28, 1171. (138) Pawlak, A.; Galeski, A. J. Polym. Sci., Polym. Phys. Ed. 1990, 28, 1813. (139) Robelin-Souffaché, E.; Rault, J. Macromolecules 1989, 22, 3581. (140) Magill, J. H.; Schultz, J. M.; Lin, J. S. Colloid Polym. Sci. 1987, 265, 193. (141) Toda, A.; Taguchi, K.; Hikosaka, M.; Kajioka, H. Polym. J. 2008, 40, 905. (142) Toda, A.; Okamura, M.; Taguchi, K.; Hikosaka, M.; Kajioka, H. Macromolecules 2008, 41, 2484. (143) Toda, A.; Taguchi, K.; Kajioka, H. Macromolecules 2008, 41, 7505. (144) Briber, R. M.; Khoury, F. J. Polym. Sci., Polym. Phys. Ed. 1993, 31, 1253. (145) Bassett, D. C.; Hodge, A. M. Proc. R. Soc. London 1981, A377, 61. (146) Xu, J.; Guo, B.-H.; Zhang, Z.-M.; Zhou, J.-J.; Jiang, Y.; Yan, S.; Li, L.; Wu, Q.; Chen, G.-Q.; Schultz, J. M. Macromolecules 2004, 37, 4118. (147) Schultz, J. M.; Kinloch, D. R. Polymer 1969, 10, 271. (148) Pearson, D. S.; Ver Strate, G.; von Meerwall, E.; Schilling, F. C. Macromolecules 1987, 20, 1133.

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