A cell model to describe the spherulitic growth in semicrystalline polymers.

INTRODUCTION

An accurate prediction of microstructure formation in semicrystalline polymers is of prime importance for parts made through injection molding, fiber spinning, or extrusion. The microstructure of the solidified polymer will dictate its physical and chemical properties and thus the quality of the manufactured products (1). In semicrystalline polymers, the solidification conditions have a strong influence on the evolution of the microstructure. This is often observed indirectly through the various experimental heat capacities reported in the literature or through direct microscopic examination. The heat capacity curves are not single-valued functions of the temperature but depend on the state of the polymer (crystalline or amorphous and liquid or solid) and, therefore, on the cooling history and the evolution of the microstructure (2). It has been shown that the limiting factors in the growth of crystals from quiescent melts are the kinetics of crystallization rather than thermodynamic factors only. Kinetics approaches to the crystallization process are therefore used to describe the evolution of the microstructure instead of classical phase change methods used to describe the solidification of water. for example. To date, several models have been developed to represent the evolution of the overall crystallinity with temperature (3). These models generally require the temperature field to predict the degree of crystallinity of a polymer, and various schemes have been presented to couple these models with the energy equations (4-6). Unfortunately, these models are often semi-empirical and need adjusting parameters that seldom have physical meaning. Thus, it becomes difficult to extract any information from these models other than the degree of crystallinity. These models also fail to predict the wide variety of entities formed during the solidification of polymers, and they neglect important mechanisms of phase transformation, such as the transport of the noncrystallizing species or the impact of the cooling rate.

It has been postulated that the phenomenon of impurity segregation in solidifying polymers is directly responsible for the spherulitic morphology commonly encountered in semicrystalline polymers (7-12). Clearly, the crystallinity of a polymer spherulite is at least partially controlled by the diffusion of the noncrystallizing species since a spherulite has always been found with some noncrystallizing material in and around its structure. This segregation of impurities in polymers was demonstrated by Barnes et al. (13), who showed that a wave of noncrystallizing species is pushed ahead of a growing spherulite boundary. Also, Moyer and Ochs (14) have shown that impurities gather between spherulite boundaries of three different polymer systems. Keith and Padden (11) demonstrated this phenomenon for polypropylene, observing heavy concentrations of impurity molecules between the spherulites and to a lesser extent along their radii. More recently, Calvert and Ryan (15, 16) and Billingham et al. (17, 18) studied the redistribution of fluorescent impurities in crystallizing polypropylene melts. Most noticeably, they studied the diffusion of atactic or amorphous fractions (which cannot crystallize because of the random configuration adopted by side groups) in isotactic polypropylene and showed that there is a significant non-uniformity of atactic content (18).

Evidence that atactic/isotactic blends behave similarly to alloys has been given by Martuscelli et al. (19). Figure 1a reproduces the experimental results of Martuscelli et al. for atactic/isotactic blends of polystyrene. In this Figure, the melting point, on the y-axis, exhibits a considerable drop with an increase in the concentration of atactic material. Furthermore, as mentioned earlier, large amounts of solute can accumulate at the boundaries of spherulites. Figure 1b is a schematic based on the measurements of Billingham et al. (18) of the distribution of atactic material in a spherulite. The measured amount of labeled (fluorescent) atactic polypropylene (LAPP) inside and around a spherulite is plotted. These two Figures illustrate and endorse that the accumulation of solute certainly plays an important role in the evolution of the spherulite's microstructure.

It is possible to relate the evolution of the microstructure directly to the transport of the noncrystallizing species by adopting concepts used in metallurgy for the treatment of alloys, with certain modifications. These concepts allow significant refinements in modeling the evolution of the spherulitic morphology. The present work introduces a cell model based on such concepts. This model describes the evolution of the spherulitic microstructure during the solidification process with the assumption that the noncrystallizing species plays a significant role in determining the solidification path of the polymer. The transport of the noncrystallizing species (antioxidants, atactic polymer, low-molecular-weight chains, etc.) is accounted for by considering diffusion in the molten polymer surrounding each spherulite. This is coupled to macroscopic heat flow calculations to obtain an accurate description of the evolution of the temperature and morphology of a polymer part.

In this paper, first the details of spherulitic growth in polymer melts are summarized. The governing equations of the model are then presented and nondimensional parameters are identified. An integral technique used to solve the equations is presented. The model calculates the evolution of the microstructure under prescribed heat flow conditions or prescribed cooling rate and isothermal crystallization. A number of associated effects such as the transport of the noncrystallizing species and the development of the interdendritic region, as reported in experiments, are illustrated.

SPHERULITIC GROWTH IN POLYMER MELTS

Crystallizable polymers are characterized by linear thread-like molecules found either organized in highly ordered molecular arrangements (crystals) or in a disordered, amorphous phase. The most common entities encountered during the solidification of a quiescent semicrystalline polymer melt are spherulites. Spherulites are polycrystalline aggregates formed from a radiating array of crystalline fibers that branch regularly to create a three-dimensional structure of approximate radial symmetry. Spherulitic growth occurs in minerals crystallizing from viscous magma, in polymers crystallizing from the melt, and in organic compound crystallizing from the melt with added thickeners (20).

The hierarchical organization of the spherulitic structure in semicrystalline polymers is shown in Fig. 2 at successive levels. At the microscopic level ([approximately]100[[micro]meter]), spherulites appear to be polycrystalline aggregates formed of crystalline fibers. At a finer scale, each branch of the spherulites consists of lamellae, the building blocks of most types of polymeric crystalline structures. Stacks of lamellae, interconnected by regions of amorphous material, grow in concert to form a branch. It was realized in the early sixties that the lamellae emerge from a peculiar arrangement of the polymer chains. The polymer chains fold back on themselves repetitively at each lamella surface, a phenomenon now known to be widespread and coined chainfolding (21). The main features of lamella organization have been the subject of numerous reviews, and the reader can refer to Bassett (22) for example.

The growth of the spherulitic morphology is still not fully understood, but it is known that their radius often varies linearly with time under isothermal conditions, which points toward a kinetics controlled growth. The work of Keith and Padden (10-12) provided the first phenomenological theory of spherulitic growth. Their theory has gained widespread acceptance, although it is now being updated in light of new experimental evidence from Bassett (23). Keith and Padden observed that spherulites are formed from a radiating and branching microstructure and pointed out that these branches are different from the dendritic morphology encountered in metals. The branching in spherulites is noncrystallographic and occurs at low angle. Dendrites, on the other hand, are single crystals for which the branching is governed by the crystalline structure. Furthermore, they have studied the growth of spherulites from polypropylene melts and observed that spherulite-forming melts have the character of alloys instead of that of pure substances. Even in the case of homopolymers, there are polydisperse components of the melt that crystallize less readily than the major component and which will be rejected from the growing solid; this phenomenon is called fractionation of the melt (the experimental evidence to this effect has been discussed in the previous section and an example was shown in [ILLUSTRATION FOR FIGURE 1 OMITTED]). The component rejected from the system is often called the "solute." The growth process creates an excess concentration of impurity or solute, which is pushed ahead of the solid-liquid interface.

The layer of solute pushed ahead of the interface has a dual effect on crystallization. First, it depresses the equilibrium crystallization point, thereby reducing the crystallization driving force. Secondly, it reduces, the growth rate due to a shortage of crystallizable material. Additionally, it has been postulated by Keith and Padden that the thickness of the boundary layer of solute determines the scale of the spherulite's branches; however, this theory is disputed by Bassett and Vaughan (9).

The extent of solute diffusion and segregation, along with the degree of undercooling, plays a crucial role in determining the overall microscopic texture of spherulites (10-12). For example, in isothermal crystallization experiments, at low undercooling and large solute content, spherulites develop a coarse and open morphology, while at large undercooling, the morphology is dense and fine. Some possible spherulitic morphologies are shown in Fig. 3. The top entity is representative of the spherulite formed at low temperature or high undercooling, while the bottom one is often encountered at low undercooling. An additional consideration for non-isothermal conditions is the cooling rate, which plays an important role since the spherulites will exhibit the various stages of the growth observed in isothermal experiments. A model that will account for these phenomena is presented below.

CELL MODEL

Formation of a spherulite can be perceived as a network of branches that bifurcate successively to create a random structure within the liquid polymer. As the spherulite grows, the volume fraction it occupies, [f.sub.g], increases at a rate controlled by the kinetics of crystallization. This volume, however, is not necessarily fully crystalline (or solid) since branches can develop with a significant amount of undercooled melt in between, creating an interdendritic region. The overall solid fraction of the spherulite can then be described with [f.sub.s] = [f.sub.i][f.sub.g], where [f.sub.i] describes this internal solid fraction of a spherulite and [f.sub.s] is the volume fraction that would be occupied by a completely crystalline spherulite. During the growth process, [f.sub.g] changes from 0 to 1. It is possible to assume that [f.sub.i] is a constant value as per Dustin and Kurz (24) for the equiaxed dendritic solidification of metals. However, it is also possible to assume that [f.sub.i] changes during the growth process and that its evolution is tied to concentration of solute around the spherulite.

The evolution of the spherulitic microstructure can be represented by a cell divided into three zones: a solid zone, an interdendritic zone, and a liquid zone, as shown in Fig. 4. A similar model has been adopted by Rappaz and Thevoz (25, 26) to describe the evolution of equiaxed dendritic growth in metal castings.

The growth of spherulites is studied by considering a spherical volume element at a uniform temperature. Each spherulite can grow to a maximum size [R.sub.tot]. [R.sub.tot] is determined by dividing the considered volume with the number of nuclei. During the growth process, a liquid zone (L) in which solute diffusion is considered surrounds the spherulite, starting at r = [R.sub.g] and ending at r = [R.sub.tot]. Between the solid (S) and liquid zone (L), there is an interdendritic zone (S + L) beginning at r = [R.sub.s] and ending at r = [R.sub.g]. In the interdendritic zone, complete mixing of the solute with the crystal branches is assumed. No solid diffusion is considered in the solid phase. Additionally, it is assumed that the impurities are well dissolved and that there is no clustering.

Symmetry boundary condition is applied on the external spherical envelope of the cell to simulate a closed system, i.e.,

[Mathematical Expression Omitted]

The symmetry boundary condition warrants some comments since some severe limitations can be foreseen. First, it cannot take into account phenomena such as triple meeting points of the solute layers or the regions growing without interference from other spherulites. The limits of such a boundary condition can be visualized with the help of Fig. 5, where combinations of spherulites are viewed from above. In (a), the appropriate boundary condition is c = [c.sub.o] at r [approaches] [infinity], while in (h), [Delta]c/[Delta]r = 0 at r = [R.sub.tot] appears to be a fairly realistic situation. However, in the intermediate configurations shown from (b) to (f), combinations of the two boundary conditions are present. A second limitation observed from this Figure is that the final spherical envelope used in this cell model does not account for the impingement of the spherulites during the growth process. It may be possible to treat this phenomenon by assuming that the final envelope of the spherulite at r = [R.sub.tot], is nonspherical, and using a shape that represents, on the average, the final microstructure of the impinged spherulites. Kelvin's tetrakaidecahedron, which is the ideal minimal-energy equivolume cell shape, can represent this geometry (27). Such a final envelope for the spherulite would improve the accuracy of the final growth process. Unfortunately, this geometry significantly complicates the calculations, and it will not be examined here. Another approach would be to treat the impingement process with the correction of Kolmogoroff (28), Johnson and Mehl (29), and Avrami (30). They compensated for grain impingement by using the extended-volume concept for the free growth of spheres distributed randomly. However, to keep the model simple and since we are dealing with only one spherulite for the time being, the Avrami correction was not implemented.

In the coming sections, each of the governing equations of the cell model are introduced and described.

Crystallization Kinetics

The growth of the interdendritic region corresponds to the growth of the tips of the branches of a spherulite. Therefore, it follows the kinetics of attachment of polymer chains given by the Lauritzen and Hoffman theory (21, 31). The growth rate of the crystal can be represented in a general manner by

[Delta][R.sub.g]/[Delta]t = [G.sub.o]exp[- [U.sup.*]/R(T - [T.sub.[infinity]]) exp[- [K.sub.g]/T[Delta]Tf] (1)

The first exponential term represents the temperature dependence of the segmental jump rate in the polymer, while the second term is the contribution from the net rate of nuclei formation on the surface of a lamella. The crystallization process arises from two contributions: the formation of a nucleus on a substrate and the pulling of a chain into the crystal. [R.sub.g] is the grain radius, t is the time, [U.sup.*] is the activation energy, R is the universal gas constant, [T.sub.[infinity]] is the temperature at which all motions associated with viscous flow cease, [K.sub.g] is a constant related to the growth regime, T is the temperature, [Delta]T is the undercooling, and f is a correction factor that accounts for the change in value of the heat of fusion at low temperatures (close to [T.sub.g]). Also, [G.sub.o] is a pre-exponential constant that gathers terms not strongly dependent on the temperature and, since the growth process can change from regime I to regime II, depending on the temperature, [G.sub.o] will change accordingly.

For the case of polymer systems with significant solute content, it is acceptable to assume that the growth rate is proportional to the concentration of crystallizable units. An entropic contribution to the free energy needs also to be added to Eq 1 (32). Since this term is negligible at low solute content, it was not added to the kinetics.

Equation 1 can be rendered dimensionless with the help of the maximum growth rate found in regime II, which is the regime prevalent throughout most of the cooling processes observed here. The following form is obtained:

[Mathematical Expression Omitted]

where

[[Lambda].sub.1] = exp[[U.sup.*]/R([T.sub.max] - [T.sub.[infinity]])] exp[[K.sub.g]/[T.sub.max][Delta][T.sub.max]f] (3)

[T.sub.max] is the temperature that gives the maximum growth rate under regime II, and [Mathematical Expression Omitted].

To account for the depletion of crystallizable material ahead of the growing interface due to the increase in solute content in the boundary layer, the term (1 -[c.sup.*]/100) multiplies Eq 2.

Heat Balance

It is possible to write a heat balance for the cell with the assumption of uniform temperature at a given time and the presence of a heat generation term due to the transformation of the solid fraction. The heat balance is then written as:

[Mathematical Expression Omitted]

where q is the external heat flow, [H.sub.f] the volumic heat of fusion, [f.sub.s] the solid fraction, and [C.sub.p] the heat capacity. The following dimensionless equation is obtained from Eq 4.

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted], [Mathematical Expression Omitted] and St = [H.sub.f]/([Rho][C.sub.p][Delta][T.sub.ref]). An equivalent heat transfer coefficient can be introduced as [h.sub.eq] = q/[Delta][T.sub.ref] and the Biot number for the cell, based on the total radius, is then Bi = ([h.sub.eq][R.sub.tot])/[k.sub.cond]. The Biot number is the ratio of the equivalent heat transfer coefficient for the prescribed heat flow at the surface of the control volume to the specific conductance of the polymer. The assumption of uniform temperature within the cell is therefore valid. at low Biot number, i.e., if the specific conductance is much larger than the heat transfer coefficient (3Bi [less than] 0.1). The Stefan number is the ratio of the latent to the sensible heat content (latent heat times the reference temperature difference). A large Stefan's number signifies that the heat released during the phase transition is absorbed very slowly by the material as a result of a variation of the sensible heat content (4).

Furthermore, since from an equilibrium phase diagram the change in temperature is related to the change in concentration,

[Delta]T/[Delta]t = m [Delta][c.sup.*]/[Delta]t (6)

where m is the slope of the line describing the change in temperature with the concentration of solute and [c.sup.*] is the concentration of the interdendritic region.

We can rewrite Eq 4 as

qS/V = - [H.sub.f][Delta][f.sub.s]/[Delta]t + [Rho][C.sub.p]m [Delta][c.sup.*]/[Delta]t (7)

where S and V are the surface and volume of the cell respectively. The dimensionless form of Eq 7 is

[Mathematical Expression Omitted]

since

[Mathematical Expression Omitted]

where [[Lambda].sub.2] = m/[Delta]T and c are commonly reported in % mass or volume fractions; it is therefore already dimensionless. [c.sub.o] is the initial concentration of noncrystallizing species and [[Lambda].sub.2] is a dimensionless parameter relating the cooling rate to the interdendritic concentration.

Since two different time scales can be used, [Mathematical Expression Omitted] or [Mathematical Expression Omitted], Eq 8 can be rewritten as

[Mathematical Expression Omitted]

where [[Lambda].sub.3] = 3Bi/Le and Le is the Lewis number (D/[Alpha]), the ratio of the mass and thermal diffusivities.

Diffusion Equation in the Liquid Zone

The diffusion of solute in the liquid region is controlled by the diffusion equation written in spherical coordinates:

1 [[Delta].sup.2]/r[Delta][r.sup.2] (cr) = 1/D [Delta]c/[Delta]t (11)

for [R.sub.g] [less than or equal to] r [less than or equal to] [R.sub.tot], where D is the mass diffusion coefficient and c(r, t) the concentration profile of solute. The initial condition for the solute profile was simply obtained from the steady-state solution of the diffusion field around a sphere, which is given in Appendix A.

The dimensionless form of the diffusion equation is

[Mathematical Expression Omitted]

with the boundary conditions

[Mathematical Expression Omitted]

Solute Balance

The symmetry boundary condition at r = [R.sub.tot] implies that solute has to be conserved within the cell. If it is assumed that there is a complete mixing within the interdendritic liquid, the solute balance of the cell is then expressed in radial coordinates,

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where [R.sub.g] is the radius of the grain or spherulite, [R.sub.s] the radius of the solid zone, [c.sub.o] the initial concentration, [c.sup.*] the concentration of the interdendritic region, anti k the partition coefficient (= [c.sub.s]/[c.sub.l]). The partition coefficient controls the amount of solute incorporated or trapped within the spherulite.

The dimensionless form of Eq 14 is

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the volume fraction of the whole spherulite and [Mathematical Expression Omitted] is the volume fraction of the solid zone only, i.e., the amount of material that can be solidified under a prescribed set of heat transfer conditions.

SOLUTION TECHNIQUES

Three important cases can be identified for Eqs 2, 10, 12, and 15 presented above. Referring to Eq 10, a first case appears when the heat flow (q or [[Lambda].sub.3]) is known or prescribed. Such a situation arises if this model was included in a large-scale simulation of the solidification process, where the heat flow is prescribed by the macroscopic simulation and serves as input to the microscopic model (33). A second case arises when the cooling rate ([Delta]T/[Delta]t or m[Delta][c.sup.*]/[Delta]t) is prescribed in Eq 10. This situation is often encountered in differential scanning calorimetry (DSC) studies of nonisothermal crystallization (34). The last case, isothermal crystallization, is commonly encountered again in DSC studies of polymer crystallization, and since the temperature is fixed, [Delta]T/[Delta]t = 0.

The unknowns to be found in order to describe the evolution of the microstructure are [f.sub.s], [f.sub.g], the solute concentration profile in the liquid zone, and [c.sup.*] in the prescribed heat flow situation or [[Lambda].sub.3] in the prescribed cooling rate or isothermal cases. All the equations are coupled and need to be solved simultaneously. Rappaz and Thevoz (25) solved a similar set of equations using finite differences. Unfortunately, this technique is very computer intensive and is not useful for practical purposes. Soon afterward, they presented a model to solve the equations by making a series of simplifying assumptions (26). A significant assumption was that the profile in the liquid zone is linear, thus arriving at a second form of the solute balance equation that allowed the system to be solved. It is, however, possible to solve the governing equations with the help of an approximate integral technique and a simple time-stepping scheme. It does not require any approximation other than the shape of the solute profile. The algorithm for each of three possible cases is presented below, starting with the more complicated prescribed heat flow situation.

Integral Method for Prescribed Heat Flow

The prescribed heat flow situation is important to consider if the cell model is to be used, as mentioned above, along with macroscopic heat flow calculations to evaluate the microstructure throughout a large polymer sample. Techniques such a finite differences, finite elements, or boundary elements may be used to perform the macroscopic calculations. The integral method used to solve the equations is summarized below, and a detailed presentation is given in Appendix B.

Solution of Eqs 2, 10, 12, and 15 presented previously requires a time-stepping scheme. The growth process can be divided in three distinct phases shown in Fig. 6. The first phase, the free growth, occurs when the solute layer [Delta] has not yet reach the radius [R.sub.tot]. Then, when [Mathematical Expression Omitted], the solute layer increases gradually in volume until [Mathematical Expression Omitted] and this is the solute buildup process. The final solidification stage, called here the secondary crystallization, is simpler since it involves only a solid zone and a mushy zone. The solution technique presented for each of those phases is summarized below, and details are in Appendix B.

Free Growth Process

The so-called free-growth regime of the spherulitic growth occurs immediately after nucleation and persists as long as the solute layers do not interact. The first value that can be computed from the equations is the growth rate at the tip of each dendrite. At a given time [Mathematical Expression Omitted], [Mathematical Expression Omitted] is first computed with the growth speed given by Eq 2 representing the kinetics of attachment of polymer chains. [f.sub.g] is now known and [f.sub.s] can now be computed.

A solute balance, written at time [Mathematical Expression Omitted] gives

[Mathematical Expression Omitted]

where [C[prime].sub.s] and [C[prime].sub.i] are the total solute content of the solid zone and liquid zone respectively, [c.sup.*] the solute concentration of the interdendritic zone, [Mathematical Expression Omitted] the total volume of the cell (4[Pi]/3), and [c.sub.o] the initial concentration, Primes are used to indicate that the variables are taken at [Mathematical Expression Omitted]. Equation 16 can be used to find [Mathematical Expression Omitted]; however, [C[prime].sub.s] and [C[prime].sub.l] are not known.

Since a time-stepping scheme is used, it is possible to compute the solid solute content with the simple trapezoidal rule as:

[Mathematical Expression Omitted]

where d[f.sub.s] = [f[prime].sub.s] - [f.sub.s]. The solute content in the solid domain is expected to vary smoothly, and Eq 17 gives an accuracy of the order of [Mathematical Expression Omitted].

The solute content of the liquid domain is computed with

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the solute distribution in the liquid zone approximated with a function of the form of a [Mathematical Expression Omitted].

If Eqs 17 and 18 are substituted into Eq 16, [f[prime].sub.s] can be isolated. [f[prime].sub.s] can in turn be substituted into the heat balance equation, and [c.sup.*][prime] is obtained by solving a quadratic expression.

The so-called penetration depth approach from Arpaci and Larsen (35) is then used to determine the solute profile in the liquid zone. This procedure is valid as long as [Mathematical Expression Omitted]. A second order profile can represent satisfactorily the solute profile in the liquid zone, and the diffusion equation is integrated to find [Delta], the length of the boundary layer of solute ahead of the growing interdendritic domain. Polynomials of higher order could of course be used.

Solute Buildup Process

Once the tip of the solute boundary layer reaches [R.sub.tot] (or [Mathematical Expression Omitted]), a technique called the volumetric rise (35) is used to account for the solute buildup at the boundary. This technique is very similar to the penetration depth approach described above. It begins by assuming a polynomial profile for the solute layer, and the diffusion equation must be satisfied between [R.sub.g] and [R.sub.tot]. The growth of [R.sub.g] is given by the crystallization kinetics, and the solute balance is used again to find [f[prime].sub.s] and [c.sup.*][prime].

The initial condition for the solute profile is given by matching the profile obtained from the penetration depth approach with the profile given by the volumetric rise method.

Secondary Crystallization

The last stage in the solidification of the grain occurs when [Mathematical Expression Omitted] i.e., when [f.sub.g] [approximately equal to] 1 but [f.sub.s] [less than] 1 still. At this point, there is only an interdendritic zone and a solid zone. The simplified mass balance equation is used again to find [f.sub.s] and the heat balance gives [c.sup.*][prime].

Time-Stepping Scheme

A simple time-stepping scheme is used to implement the equations for each stage of the growth process. This scheme is represented in Fig. 7. It consists essentially in i) moving the tip of the dendrite with the growth expression; ii) computing the new interdendritic concentration [c.sup.*][prime]; iii) finding the change in temperature [Delta]T; iv) computing the change in solid fraction [f.sub.s]; v) computing the new boundary layer thickness [Delta]. This scheme varies slightly of course depending on which phase of the growth process is being considered.

Prescribed Cooling Rate

A prescribed cooling rate ([Delta]T/[Delta]t = const.) is often used in DSC experiments of polymeric materials to obtain heat capacities and study non-isothermal crystallization kinetics. The solution of the prescribed cooling rate case is simpler than the preceding one since it does not require to find [c.sup.*]. The relationship

[Mathematical Expression Omitted]

allows us to find [c.sup.*][prime] with

[Mathematical Expression Omitted]

which is exact for a linear cooling rate.

The same integral technique presented in the previous section can then be used to solve for [f.sub.s], [f.sub.g] and the solute profile in the undercooled melt.

Isothermal Crystallization

The isothermal crystallization process is treated quite similarly to the prescribed cooling rate case. Again, in the context of Eq 10, we can write

[Mathematical Expression Omitted]

since [Mathematical Expression Omitted]. [c.sup.*] is then fixed.

For a given [Mathematical Expression Omitted], [c.sup.*] is initially found with

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the isothermal crystallization temperature. The time-stepping scheme then follows again the same procedure outlined in the prescribed cooling rate section.

Computation of the Final Crystallinity

Hoffman et al. (21) give an interesting discussion on the physical significance of the degree of crystallinity often used by investigators working with polymers. This degree of crystallinity is commonly calculated from thermodynamic expressions such as

[Mathematical Expression Omitted]

where H is the enthalpy, [Mathematical Expression Omitted] is the specific volume, and the subscripts s, l, and c refer to the sample, the liquid phase, and the pure crystalline phase respectively.

The amorphous content of the polymer being studied, 1 - [Chi], measured with the above relationships, lumps the amorphous fractions coming from different origins. 1 - [Chi] is a measure of the total amount of amorphous material in the polymer and 10-15% of this noncrystalline material is related to the properties of the fold surface of each lamella. The truly noncrystalline or liquid component begins to appear when [Chi] falls below 0.90 or 0.85. Furthermore, the maximum degree of crystallinity attainable in practice decreases greatly with an increase in the molecular weight. For a molecular weight of about [10.sup.4]-[10.sup.5], the maximum degree of crystallinity is 80-90%, but it is reduced to 40% when the molecular weight is above [10.sup.6]. This decrease in crystallinity is associated, according to Hoffman et al. (21), with an increase of the interlamellar links and the irresolvable entanglements of polymer chains.

From this model, it is possible to compute the contribution of the atactic material or low-molecular-weight component dispersed in the melt (the solute) to the amorphous content (1 - [Chi]) of a spherulite. The amount of solute incorporated in a spherulite is controlled by the partition coefficient k. The computed [C.sub.s], obtained at the end of the simulation, gives the solute contained in the interlamellar regions and trapped between the branches. The amount of solute trapped in the outer sphere surrounding the spherulite is simply given by V[c.sub.o] - [C.sub.s]. However, to determine with the equations presented above the crystallinity of a solidified spherulite, it is required to compute the interlamellar spacing and account for the molecular weight effects. These considerations could be added subsequently to the model. The computations presented below should be accurate for highly linear molecules of average molecular weight, such as polyethylene.

RESULTS AND DISCUSSION

Polyethylene (PE) is the material chosen to evaluate the model owing to the availability of most of its material and thermodynamic properties and its high degree of crystallinity. The kinetic data used in Eq 2 are presented in Table I and were obtained from Hoffman et al. (21). The thermodynamic data were obtained from Quirk and Alsamarraie (36) and are also presented in this table.

The initial conditions to be provided to the model are the nuclei size ([R.sub.g]), the initial undercooling ([Delta][T.sub.u]), the uniform initial concentration of impurities ([c.sub.o]), and the heat flow (or cooling rate). Typical initial parameters that would be given by experimental observation are [R.sub.tot] = [10.sup.-3] cm, [Mathematical Expression Omitted], and [c.sub.o] = 5% of solute (low-molecular-weight molecules here). An undercooling of [Mathematical Expression Omitted] is common for the apparition of the first nuclei for a melt cooled at a rate of -20 K/min, or for the corresponding heat flow determined with

q = [Rho][C.sub.p] [R.sub.tot]/3 [Delta]T/[Delta]t (24)

This set of values is chosen as a reference state for the three possible cases presented below, i.e., prescribed heat flow, prescribed cooling rate, and isothermal solidification.

[TABULAR DATA FOR TABLE 1 OMITTED]

Prescribed Heat Flow

There are five dimensionless parameters governing the behavior of the system of equations considered. These parameters are [[Lambda].sub.1], [[Lambda].sub.2], [[Lambda].sub.3], St, and k. The values of the dimensionless parameters corresponding to the data presented in Table 1 for polyethylene are [[Lambda].sub.1] = 1.761 x [10.sup.4] in regime II at [T.sub.max] = 337K, [[Lambda].sub.2] = -0.0268, [[Lambda].sub.3] = -17.84, St = 0.5732, and k = 0.4. For these conditions, the temperature of the cell, the solid fraction, and the grain fraction computed with the time-stepping scheme are shown in Fig. 8.

Several interesting phenomena can be observed from Fig. 8. First, the recalescence phenomenon, where the temperature rises due to the latent heat release, is well illustrated and a substantial temperature plateau develops immediately after the recalescence. This plateau occurs during the quasi-steady-state growth regime of the spherulite. The rise in temperature associated with the recalescence also affects the spherulite growth rate and slows down slightly the progress of [f.sub.g], as it can be observed in Fig. 8. Since the solid fraction [f.sub.s] cannot increase beyond the limit imposed by the heat flow and since the crystallization kinetics are driving [f.sub.g] faster than [f.sub.s], a significant interdendritic region develops in the spherulite at about [Mathematical Expression Omitted] = 0.003. This interdendritic region implies that a significant amount of undercooled melt is trapped between the branches of the spherulite.

Various points on the temperature curve of the cell have been marked from A to E, and they correspond to the solute profiles shown at various times in Fig. 9. At point A, a region in which the solute content is significantly larger than the undercooled melt is rapidly formed in front of the growing spherulite, and the size of this region is given by the boundary layer thickness [Delta]. The concentration ahead of [Mathematical Expression Omitted] has increased from the initial value [c.sub.o] to a maximum of about 8%. A concentration gradient is formed from the pile-up solute in front of the growing spherulite, and this gradient induces diffusion of the solute toward the regions of low concentration. The extent of the solute diffusion process in the undercooled melt is determined by the steepness of the concentration gradients, by the diffusion coefficient, by the partition coefficient which dictates the extent of solute trapping in the spherulite and by the growth rate. The higher the growth rate, the less time the solute has for diffusing away from the growing interface ([R.sub.g]) into the under-cooled melt and more solute is trapped in the spherulite since the solute concentration [c.sup.*] is larger than for a slow moving interface.

At point B, an interdendritic region begins to form due to the difference in the growth offs and [f.sub.g]. This region develops during the recalescence plateau (point C), and the solute buildup process begins and finishes very quickly at point D. At this point, the solute layer has reached [Mathematical Expression Omitted], and the boundary layer of the spherulites are interacting with one another (solute buildup process). Due to the low mass diffusivity, the solute layer is rather small, and the transition from free-growth to solute buildup and secondary crystallization is rather abrupt. This is illustrated by a small kink in the temperature curve near point D. A significant interdendritic region has developed at this time. The solute accumulated around the spherulite forms a spherical shell of impurities, and this is illustrated by the solute profile corresponding to point E.

The concentration of solute in the solid spherulite, shown in Fig. 9 is somewhat uniform throughout the spherulite, except at its edge. It rises slightly near the center, then dips close to the midradius of the spherulite to rise again close to the boundary as the concentration in the liquid increases due to the solute pile-up on its edges. The profile in the solid domain is directly linked to the concentration of the undercooled melt by the partition coefficient. The profile of the undercooled melt, in turn, is tied to the growth process or ultimately to the cooling rate, and significantly different profiles are to be expected with different cooling conditions; i.e., a fast growing spherulite will have a small boundary layer and steep concentration gradients compared to a slow growing spherulite in which the solute is allowed to diffuse in the undercooled melt.

Prescribed Cooling Rate

A very different behavior is found when a cooling rate is prescribed on the cell, as can be observed in Fig. 10 where [f.sub.s], [f.sub.g], and the temperature profile are shown. The value of the dimensionless parameters used in this case are [[Lambda].sub.1] = 1.761 x [10.sup.4] in regime II, [[Lambda].sub.2] = -0.0268, St = 0.5732, k = 0.4, and [Mathematical Expression Omitted]. [[Lambda].sub.3] is allowed to vary since the heat flow is not prescribed. Obviously, no recalescence is observed here since the cooling rate is prescribed and the polymer is assumed to follow the imposed temperature. Also, no interdendritic region develops due to the combination of high growth rate and high cooling rate. The rapid decrease in temperature allows rapid growth regimes to be reached, which means rapid changes in [f.sub.g], and the cooling rate is enough to allow [f.sub.s] to "keep up" by removing heat sufficiently fast.

Various points on the [f.sub.s] and [f.sub.g] curves have been marked from A to D, and they correspond to the solute profiles shown in Fig. 11. These profiles in the spherulite are shown at various times during the growth process. At point A and B, the spherulite grows with no interdendritic region and the length of the solute boundary layer changes with the changing growth rate, which is a function of temperature. The maximum concentration reached is about 13%. At point C there is still no interdendritic region and the solute boundary layer is small. The solute profile increases gradually, once [Mathematical Expression Omitted], but in such a narrow region that it is barely visible on the graph.

The solute concentration profile in the nearly fully solidified spherulite, observed in Fig. 11 at point D is almost uniform rising slightly and regularly except at the outer edge, where the buildup occurs in a very narrow region. It can be noticed that the concentration profile in the solid zone is very close to [c.sub.o] (5%) due to the high concentration of solute achieved in the boundary layer. This high concentration is reached because of a combination of a high growth rate and rapid cooling, which prevents the formation of an interdendritic region. To maintain a mass balance, the boundary layer has to be very narrow since a significant amount of solute is incorporated in the spherulite.

Isothermal Crystallization

The curves of [f.sub.s] and [f.sub.g], obtained under isothermal conditions, are shown in Fig. 12 for [Mathematical Expression Omitted], which gives approximately the maximum growth rate. The values of the dimensionless parameters are identical to those of the previous section. It can be seen that the curves of the solid and grain fractions follow each other up to [f.sub.s] = 0.8, at which point they part away. This phenomenon is related to the solute buildup in the liquid domain. The solute profile increases gradually, once [Mathematical Expression Omitted], and it eventually becomes almost uniform throughout the liquid domain. Since this uniform profile is very close to [c.sup.*], the fixed interdendritic concentration, [f.sub.s] must slow its growth to satisfy the mass balance. [f.sub.s] cannot change subsequently once [f.sub.g] = 1 since [c.sup.*] is fixed.

Several points were also marked from A to D on the [f.sub.s] and [f.sub.g] curves, and the corresponding profiles can be observed in Fig. 13. It can be immediately seen that the solute profile is very well developed at point B, compared to the profile of point A. The slow moving interface allows the boundary layer to develop, and a small interdendritic region also appears. Point C illustrates a profile in the liquid domain that is not very different from B, except that the interdendritic region is somewhat larger. Point D shows the final concentration profile. This profile does not change later on since [c.sup.*] is fixed, [f.sub.g] = 1, and the mass balance must be satisfied. This can only be accomplished if [f.sub.s] does not change, which is observed here.

Limitations of the Model

A drawback of this approach to modeling the spherulitic growth is that parameters such as the partition coefficient k and the phase diagram are time-consuming to obtain. A description of the evolution of the temperature of the polymer with its solute content is required, which demands time-consuming experiments. Furthermore, to be able to compare the results with experimental data, such as those obtained from DSC studies, the model needs to be implemented in an average sense for a small control volume. Nucleation phenomena need to be accounted for, and the impingement process has to be included. It will then be possible to compare directly the computed change in enthalpy with the experimental observations, and this is the subject of a coming paper (37).

CONCLUSION

A model that couples crystallization kinetics, mass diffusion, and heat transfer to predict the evolution of the microscopic texture found during the solidification of semicrystalline polymer has been presented in this paper. It can simulate the evolution of a spherulite under a wide variety of conditions for homopolymers and polymer blends. The treatment presented can account for several important crystallization phenomena, such as the diffusion of the noncrystallizing species (or the other polymer in a blend), and it gives information related to the texture of the polymer. It is based on parameters with well-defined physical meanings. Five dimensionless quantities were identified that affect the cooling profile and the microstructure. Very useful information, such as the cooling time, morphology, and solute distribution were obtained. The development of the microstructure was; explored under conditions of prescribed heat flow, prescribed cooling rate, and isothermal crystallization. Very different results were observed in each case illustrating the strong influence of the cooling conditions on the evolution of the microstructure. The development of a significant interdendritic region was found under prescribed heat flow, along with a rise in temperature associated with the self-heating of the polymer due to the latent heat release. A much smaller interdendritic region developed under isothermal conditions. The maximum solid fraction was below unity due to the imposed conditions. No interdendritic region developed under the fixed cooling rate used since the available heat flow was sufficient to prevent any secondary crystallization.

APPENDIX A

Initial Conditions for the Free Growth Stage

The initial conditions to be specified for the free growth process include the starting temperature, concentration [c.sup.*], solid fraction [f.sub.s], and an initial boundary layer [Mathematical Expression Omitted]. All these parameters are given by experimental observation except for [Mathematical Expression Omitted]. To find [Mathematical Expression Omitted], it is possible to use the steady-state solution for the diffusion field around a sphere

[Mathematical Expression Omitted]

where the constants A, B, and C are given by

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] initially. This results in the following profile:

[Mathematical Expression Omitted]

where

[Mathematical Expression Omitted]

[Mathematical Expression Omitted] is finally found by equating the amount of solute given by the steady-state solution and the approximate profile

[Mathematical Expression Omitted]

which gives the quadratic equation for [Mathematical Expression Omitted]

[Mathematical Expression Omitted]

The governing equations can now be solved with a time-stepping scheme. The penetration depth approach is valid as long as [Mathematical Expression Omitted]. If [Mathematical Expression Omitted] however, the solute buildup process begins and we have to use another approach.

APPENDIX B

Detailed Solution Techniques

As mentioned in the main body of this paper, the growth process can be divided in three distinct phases shown in Fig. 6. Each of these phases is reviewed below.

Free Growth Process

The so-called free-growth regime of the spherulitic growth occurs immediately after nucleation and persists as long as the solute layers do not interact. The first value that can be computed from the equations is the growth rate at the tip of each dendrite. At a given time [Mathematical Expression Omitted], [Mathematical Expression Omitted] is first computed with the growth speed given by Eq 2 for the kinetics of attachment of polymer chains.

A solute balance, written at time [Mathematical Expression Omitted] is given by Eq 16. [Mathematical Expression Omitted] and [Mathematical Expression Omitted] in Eq 16 are still unknown however.

Since a time-stepping scheme is used, it is possible to express the solid solute content with Eq 17. An approximate profile is used to represent the solute distribution in the liquid zone. This profile, due to the spherical geometry considered, must have the form

[Mathematical Expression Omitted]

to obtain a good approximation, as discussed by Ozisik (38). Using such a profile, it is possible to compute the solute contained in the liquid zone as

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

Carrying out the integration gives

[Mathematical Expression Omitted]

where g(z) is a function that depends on the polynomial chosen and z is the ratio of [Mathematical Expression Omitted]. For a profile of the form

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

we have simply

g(z) = 1 + z + [z.sup.2]/4 (35)

If Eqs 33 and 17 are substituted into Eq 16 and [Mathematical Expression Omitted] is isolated, the following is obtained

[Mathematical Expression Omitted]

Equation 36 gives an expression that can be used to compute the change of solid fraction [df.sub.s] from one time-step to another, provided that [c.sup.*][prime] is known.

The heat balance equation gives the means to find [c.sup.*][prime]. To do this, it suffices to compute [df.sub.s] and [f.sub.s] - [f.sub.s] and substitute this into the heat balance given by Eq 10 The following quadratic expression for [c.sup.*][prime] is obtained

[Mathematical Expression Omitted]

It is to be noted here that z[prime] is not known. However, it appears reasonable to assume that the ratio of [Mathematical Expression Omitted] does not change considerably from one time-step to another, i.e., [Delta]z/[Delta]t [approximately equal to] 0.

The penetration depth approach (35) can now be used to model the evolution of the solute layer in the liquid zone. This procedure is valid as long as [Mathematical Expression Omitted] 1. A second order profile can represent satisfactorily the solute distribution in the liquid zone. This profile has to satisfy the diffusion equation

[Mathematical Expression Omitted]

in [Mathematical Expression Omitted] and where U = c - [c.sub.o]. The following set of boundary conditions is associated to the diffusion equation:

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted]

The diffusion equation can be integrated over the solute-layer thickness [Mathematical Expression Omitted], i.e., from [Mathematical Expression Omitted] to [Mathematical Expression Omitted], and after using the boundary conditions the following balance equation is obtained:

[Mathematical Expression Omitted]

where

[Mathematical Expression Omitted].

Since we are dealing with a spherical geometry, a profile of the form

[Mathematical Expression Omitted]

can be chosen. The various constants [a.sub.i] can be obtained from the boundary conditions. The corresponding profile is given by Eq 34. Then, [Theta] is simply [Mathematical Expression Omitted]. The solute layer thickness [Mathematical Expression Omitted] is found with the help of the balance Eq 41 and is given by the integration of the following ordinary differential equation:

[Mathematical Expression Omitted]

When [Mathematical Expression Omitted], the solute buildup process begins, and we have to use another solution technique.

Solute Buildup Process

Once the tip of the solute boundary layer reaches [Mathematical Expression Omitted], a technique called volumetric rise is used to account for the solute buildup at the boundary. This technique is very similar to the penetration depth approach described above. It begins by assuming a profile of the form given by Eq 42 and the two following boundary conditions:

[Mathematical Expression Omitted]

The profile obtained is

[Mathematical Expression Omitted]

where [a.sub.1](t) is a function of time and needs to be determined by the diffusion equation.

The solute balance given by Eq 16 is also required, and the solute content in the liquid zone is

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is given by

[Mathematical Expression Omitted]

A relationship is thus obtained for [f[prime].sub.s]:

[Mathematical Expression Omitted]

Writing [df.sub.s] as [df.sub.s] = [f[prime].sub.s] - [f.sub.s] into the heat balance equation, the following quadratic equation is obtained for [c.sup.*][prime]:

[Mathematical Expression Omitted]

[a.sub.1] is found by integrating the diffusion equation. The following differential equation needs to be integrated:

[Mathematical Expression Omitted]

The initial condition for [a.sub.1] is given by matching the profile obtained from the penetration depth approach with the profile given by the volumetric rise method. This gives

[Mathematical Expression Omitted]

Secondary Crystallization

The last stage in the solidification of the grain occurs when [Mathematical Expression Omitted], i.e., when [f.sub.g] [approximately equal to] 1 but [f.sub.s] [less than] 1 still. At this point, there is only an interdendritic zone and a solid zone. After doing a solute balance, the solid fraction at time t + [Delta]t is simply

[Mathematical Expression Omitted]

Writing again [Mathematical Expression Omitted] and substituting into the heat balance will give the following quadratic equation:

[Mathematical Expression Omitted]

ACKNOWLEDGMENTS

The financial support of the E. I. DuPont Co. and the Delaware Research Partnership Program is gratefully acknowledged. We also would like to thank Dr. J. M. Schultz for helpful discussions.

NOMENCLATURE 

Bi = Equivalent Biot number, q[R.sub.tot]/[Delta]Tk.

c = Concentration (%).

[c.sup.*] = Interdendritic concentration (%).

[Mathematical Expression Omitted] = Concentration of the solid (%).

[c.sub.o] = Initial concentration (%).

[Rho][C.sub.p] = Volumic specific heat, ([Jm.sup.-3][K.sup.-1]).

[C.sub.s] = Total solute content of the solid domain (%[m.sup.3]).

[C.sub.[Iota]] = Total solute content of the liquid domain (%[m.sup.3]).

[f.sub.g] = Grain volume fraction.

[f.sub.s] = Solid volume fraction.

D = Mass diffusivity ([cm.sup.2]/s).

[h.sub.eq] = Equivalent heat transfer coefficient (W/[m.sup.2]/K).

[H.sub.f] = Volumic latent heat (J/[m.sup.3]).

k = partition coefficient, [C.sub.s]/[C.sub.1].

Le = Lewis number, D/[Alpha].

m = Liquidus slope, dT/dc (K/% or K/kg/[m.sup.3]).

q = Cooling heat flow (W/[m.sup.2]).

[Mathematical Expression Omitted] = Dimensionless radius, r/[R.sub.tot].

R = Universal gas constant (8.3145J/molK).

[R.sub.g] = Grain radius (m).

[R.sub.s] = Solid radius (m).

[R.sub.tot] = Solid radius (m).

S = Total surface of the cell ([m.sup.2]).

St = Stefan's number, [H.sub.f]/[Rho][C.sub.p][Delta]T.

[Mathematical Expression Omitted] = Dimensionless time, [Mathematical Expression Omitted].

[Mathematical Expression Omitted] = Dimensionless time, [Mathematical Expression Omitted].

[Mathematical Expression Omitted] = Cooling rate (K/s).

T = Temperature (K).

[T.sub.[infinity]] = (K).

U = Universal exponential term of the retardation factor (J/mole).

V = Total volume of the cell ([m.sup.3]).

[Alpha] = Coefficient of heat diffusion ([m.sup.2]/s).

[Delta] = Boundary layer thickness (m).

[[Lambda].sub.1] = First dimensionless parameter, [[Lambda].sub.1] = exp [[U.sup.*]/R([T.sub.max] - [T.sub.[infinity]])] exp[[K.sub.g]/[T.sub.max][Delta][T.sub.max]f].

[Lambda].sub.2] = Second dimensionless parameter, m/[Alpha]T.

[[Lambda].sub.3] = Third dimensionless parameter, 3Bi/Le.

[Rho] = Density(g/[cm.sup.3]).

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