Znd. Eng. Chem. Res. 1995,34, 3481-3487
3481
The Neumann-Stefan Problem and Its Recent Widening as an Advanced Topic of Transport Phenomena H. Janeschitz-Kriegl,*Ewa Ratajski, and G. Eder Institute of Chemistry, Linz University, A-4040 Linz, Austria
The classical Neumann-Stefan problem (progress of crystallization from a quenched wall) can only be adapted to realistic situations if crystallization kinetics are introduced. After a review of the theoretical achievements attention is focused on the determination of the kinetic parameters which will enable us to simulate processes in isotactic polypropylene numerically, including the prediction of internal structures such as number (and size) of spherulites per unit volume and transcrystallization areas near the mold surface. Also with a proper measurement of the number of nuclei and of the growth speeds of spherulites as functions of temperature, heat transfer considerations play a key role. The influence of flow, which manifests itself in injection-molded parts by highly oriented surface layers, has separately been treated.
Introduction
distance
In the age of the computer, when numerical simulations of processes are in vogue, one easily forgets that, in many cases, the physical data for such a simulation do not exist t o a sufficient extent and, partly as a consequence, the use of more general considerations is far from superfluous. In that which follows a sketch will be given of a correct treatment of solidification processes by crystallization,replacing the classical, but inconsistent equations by Neumann (see Carslaw and Jaeger (1959))and Stefan (1891). Doing so, one recognizes a wide field of research which is only scarcely explored. In particular, the crystallization kinetics of isotactic polypropylene will be investigated. The reader will see that, also for this purpose, a series of special heat transfer problems are to be solved. The influence of flow on the crystallization kinetics will be omitted in this presentation. It has recently been treated by Liedauer et al. (1993, 1995) and published separately. It will become apparent that, in the absence of flow, the features of the process of solidification of polymer melts are very similar to those observed with metals. However, the work with polymer melts is characterized by several advantages, viz., lower crystallization temperatures, slower processes, and transparency of the melt.
The Neumann-Stefan Treatment and Its Shortcomings According t o this theory, for a slab of sufficient thickness one obtains the so-called square root law, as is well-known. This means that a crystallization front moves from the quenched wall into the melt, leaving behind a solidified layer growing with the square root of time, as counted from the moment of the quench. Besides the fact that a sudden quench is impossible for practical reasons, another very unrealistic feature adheres to this model. In fact, the assumption is made that crystallization (i.e., solidification) occurs without any delay when, during the cooling process, the thermodynamic melting point is reached. Such an approach can only be useful if the progress of the crystallization front is very slow. However, this theory predicts an infinitely fast progress at the very beginning. This means that this classical treatment is inconsistent. In Figure 1 the situation is sketched. In this figure the progress of crystallization from a quenched wall at x = 0 is plotted against time for the half-infinite space.
Neumann-Stefan
-
zone
- _ _ - - - -- - - -front
'0 time Figure 1. Schematic presentation of the progress of crystallization into the half-infinite space.
The classical result is represented by the uppermost curve. If one allows for supercooling of the crystallization front, one creates a quite different situation. Whereas in the classical model the melt in front of the crystallization area is free of nuclei per definition, one can no longer make this assumption for the real case. If the crystallization boundary is supercooled, also a zone of the melt in front of this boundary reaches temperatures below the equilibrium melting point. In this zone nuclei can spring up, causing spherulitic crystallization. In principle these spherulites have a center which, according to Schulze and Naujek (1991), is shifting in the direction of the lower temperatures, which is the reverse direction of the growth process. This should result in a lower speed of progress, but this effect is overcompensated by an enhanced growth speed of the spherulites. Recently, the third author of this review has formulated a pertinent criterion: He gave an upper bound for the temperature gradient in terms of the local number of nuclei. However, if this gradient is not too large, one may ignore this effect. In fact, the preferential direction of growth is into an area, where other spherulites have started growing earlier, so that much of the volume is already occupied. A realistic treatment of the progress of an undisturbed crystallization front has a finite initial slope (which appears to represent the growth speed of spherulites at the temperature of the quenched wall). This becomes evident if one considers the birth of such a front. Actually, one can create a wall which contains no growth centers (if covered by a liquid film or by a thin layer of a glass). From such a wall, no crystallization front can take off. A front is only formed if the wall is capable of forming nuclei and if the average distance of nuclei on the wall surface is smaller than that between nuclei in the adjacent supercooled liquid zone. In such a case, half
0888-588519512634-3481$09.00/0 0 1995 American Chemical Society
3482 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 ..
~
.. ..
~
.
gets these impressive formations. This picture clearly shows how general the validity is of the considerations just given.
Some Considerations for Samples of Finite Thickness Even if the half-time l m ofconversion (crystallization) is dependent on a greater number of factors, it seems useful to compare this time with the time t ~ which . is needed for thermal equilibration. This latter time reads
where D is the (finite) thickness of the considered slab of infinite lateral extension, a is the heat diffusivity of the melt and St is the Stefan number given by:
Figure 2. Icelandic rock formation, clearly showing the columnm phase (transcrystallizationwith polymers). Courtesy of Elisabeth Liedauer.
spherulites start to grow at the surface. After some time they impinge on the surface. Afterward they can only grow in a direction perpendicular to the said surface, forming the well-known columnar phase of metallurgy or, according to Wunderlich (1973). the “transcrystallized” layer in polymers. In Figure 1 the progress of this transcrystallization is shown by the lower (straight) line. In principle, however, such a front must be superseded earlier or later by a diffuse crystallization zone, as caused by nucleation in the zone of supercooled melt. The reason for this supersession is that such a zone always moves faster. Whereas the front can be compared with the advance of a single runner on a sports field, the zone is comparable with the result of a relay race where every time the baton is handed over, two arm lengths are gained. These two arm lengths are the radii of impinging spherulites of which one is nucleated in front of the other. Thus, one may notice that the structure of the solidifymg material is completely governed by the nucleation and growth mechanisms. In fact, the Neumann-Stefan model is not only inconsistent, it is also unable to predict any structure. Out of the idea that crystallization is a self-enhancing process (at least in the beginning), Malkin et al. (1984) created a model in which crystallization is treated like a chemical reaction. For the heat transfer problem, such a model certainly represents a n improvement. But there are still two objections. One is that the crystallization speed is certainly not a unique function of the momentary degree of crystallinity: In reality it depends on the number of spherulites over which the volume fraction of crystallinity is distributed a t the moment of observation. (Many small spherulites of a certain total volume have a much larger growth surface than a few big spherulites of the same volume.) The second objection is that also Malkin’s model is unable to predict a structure. In Figure 2 a picture is given of volcanic rocks in Iceland. On this picture the columnar phase is clearly seen. Because of the fact that, by shrinkage during cooling, the columns are separated from each other, one
where Ti and T, are the initial melt temperature and the temperature of the quenched wall, c is the heat capacity of the melt, and h is the latent heat of crystallization per unit mass. The denominator gives the amount of heat to be removed per unit mass, if no crystallization takes place. This amount is removed during the time D2/4u. The relative increase of this time as a consequence of crystallization is given by (1 + St). According to Berger and Schneider (1986) the ratio
R = tdtln
(3)
is of great importance. It characterizes the type of process occumng. For a small value ofR the sample is cooled down before a noticeable crystallization sets in (thin slab, slow crystallization). Such a process is called a nucleation rate controlled process. For a large value of R one obtains almost the classical situation, viz., a (narrow, frontlike) zone moving from the surface into the sample (thick slab, fast crystallization). Such a process is called a (heat) diffusion controlled process. Examples for St 3 0 were given by Berger and Schneider (19861, whereas Astarita and Kenny (1987) treated a process with St B 1, which seems more relevant for metals. The latter authors proposed to call the number R after the first author of the present paper, but W. Schneider’s name is also eligible. At the time when these papers were published, however, no detailed knowledge on the kinetics was available, so that only an overall crystallization rate as a function of temperature was assumed, going to zero at the melting point and at the glass transition temperature and showing a maximum halfway between these temperatures.
The General Theory This general theory was initiated by Schneider et al. (1988). The idea of these authors was to replace the kinetic equations, as first given by Kolmogoroff (1937). Avrami (1939,1940,1941), and others, by more suitable equations. In fact, the just mentioned equations are of the integral type, whereas the equation of heat conduction is a differential equation. Schneider and coworkers created a set of differential equations (rate equations) for the kinetics by differentiating the mentioned integrals step by step with respect to time. At every step a useful auxiliary function is created. After
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3483 the first step the corresponding auxiliary function turns out to be the internal growth surface, enabling a correct treatment of the progress of crystallization. (As with Kolmogoroff and the other mentioned authors, only a sort of primary crystallization, a geometric filling up of space by impinging spherulites of constant specific volume was meant.) After the third differentiation the number of nuclei per unit volume, which has sprung up until this moment, emerges. After an extra, fourth differentiation one obtains the rate of nucleation. (A volume change by shrinkage can be taken into account by a quasi-steady growth treatment, comparable with the steady flow treatment of a fluid entering a conical tube of only a slight taper.) These equations only hold for bulk nucleation. Eder (1995)extended this treatment t o a description of the influences of confining walls (transcrystallization etc.). For that purpose he introduced the mathematical tool of the Poisson (point) processes. One of the results of this treatment is remarkable for those who are familiar with Avrami’s equations and, in particular, with the Avrami index: Noninteger indices can be explained in this way, varying from less than 2 t o more than 3,if for the bulk an index of exactly 3 is predicted (constant number of nuclei created at time zero, three-dimensional growth). Finally, some words have t o be added with respect t o the time parameter quoted. Mostly, kinetic parameters are functions of temperature. If the (local)temperatures within the sample change with time according t o the heat transfer problem, quantities like nucleation rate or growth rate become functions of time indirectly. In this sense the mentioned differentiations with respect t o time furnish the correct auxiliary functions also for non-isothermal situations, if the local temperature gradient is tolerable (see The Neumann-Stefan Treatment and Its Shortcomings). The coupling between the rate equations and the equation of heat conduction occurs through the (gradually) evolved heat of crystallization.
Troubles with the Kinetic Data Even if a treatment of far-reaching correctness is now available, this does not mean that it can be applied to practical problems. According to Eder et al. (1990)the most general presentation of the nucleation rate may be that by an activation time spectrum, viz.,
where Ni is the number of potential sites per unit volume for nucleation process “i”, Ti is the corresponding activation time as a function of temperature, and T is the temperature as a function of time. Remarkably, in this expression time appears also explicitly in the upper bound of the integral. This equation, however, contains too many parameters, which never can be determined by experiment. (Compare this with the difficulties encountered in the determination of a relaxation time spectrum for linear viscoelasticity, where mechanical measurements are by far superior to thermal measurements.) There are two alternative simplifications. One of these simplifications consists of the assumption that there is only a single mechanism (cf. Avrami). An additional assumption, as made by Nakamura et al.
(19721,is characterized by the so-called isokinetic case, where the frequency with which new nuclei spring up and the speed of growth G(T) obey the same temperature dependence. In other words, the product G(T>d r ) = A
(5)
should be constant. However, Eder (1995)has given a formal proof that the influence of the thermal history on the internal structure of the final product is wiped out by this assumption. This is in contrast to the experience that fine-grained structures can be obtained by particularly fast cooling. Moreover, if it is true that many polymers of industrial interest show an Avrami index of 3 at low degrees of supercooling (see, e.g., Mandelkern (1964)),where the crystallization process is slow enough for practical experimentation, one can also forget an increase of the number of spherulites (decrease of their size) with enhanced cooling rates. In fact, Avrami index 3 means that all potential nucleation sites are activated simultaneously and spherulites grow from the onset. Thus, there remains only the announced alternative way of simplifying eq 4 and retaining, a t the same time, the ability of predicting an influence of thermal history on the finally obtained structure. For this purpose the assumption is made that the activation times of all processes, which are certainly infinite at the equilibrium melting point, remain infinite during cooling until their special activation temperature T,is reached. At this temperature the corresponding activation time ti is assumed to jump t o zero. In this way one obtains for the total number of activated sites per unit volume
In this way one obtains not only G(T) as a unique function of temperature but also N(T)as such a function. Formally, one obtains also a(t)= cW(T(t))/dtas the rate of activation. However, this value of a(t)is not needed in our approach, because one can easily omit the last step of differentiation, as carried out by Schneider et al. As a consequence, one step of integration becomes superfluous by this thoroughly deterministic approach. As we shall see in the following sections, eq 6 seems realistic at least for isotactic polypropylene. At the same time we shall get a feeling for the scanty chances of a more detailed evaluation.
The Number of Nuclei as a Function of Temperature For an industrial polymer like isotactic polypropylene it becomes technically impossible to carry out a sufficiently fast and well-defined quench t o a chosen level of temperature of crystallization. In fact, the material starts crystallizing already during the cooling procedure. With this polymer one is rather far away from the limiting case of a purely nucleation rate controlled process. As a remedy, we found the application of the differential scanning calorimetry (DSC) (see Eder and Janeschitz-Kriegl(1993),Janeschitz-Kriegl et al. (19931, Eder et al. (19931,and Janeschitz-Kriegl (1994)). In fact, we made use of the control system of this machine in creating a constant cooling rate. We could show that such a cooling rate exists not only in the furnace, but also in the center of the sample up to the moment that crystallization sets in. It was shown by our group that
3484 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995
IO0
105
1 IO
I15
I20
125
Crystallization Temperature ["C]
Figure 3. Number of spherulites per unit volume against true crystallization temperature for isotactic polypropylene Daplen KS 10.
one has to introduce an effective heat transfer coefficient between the center of the sample and the furnace. The usefulness of such a heat transfer coefficient was also checked by numerical simulation. So far, it has always been assumed that the observed broadening of the DSC peak and its shift to lower temperatures with increasing cooling rate was due to the changing crystallization kinetics. However, we have been able to show that the broadening must mainly be ascribed to the insufficient heat transfer. After the application of the necessary corrections a peak, as observed at a high cooling rate, becomes almost as narrow as a peak observed a t a low cooling rate. Also the shift on the temperature axis is reduced. (At a cooling rate of 100 Wmin the peak of the corrected curve is found at a temperature of almost 20 "C above the corresponding temperature of the uncorrected curve!) Thus, the influence of the cooling rate on the effective crystallization temperature is less than has been assumed so far. On the other hand, a contracted peak gives a much better idea of the value of the effective crystallization temperature. Heed should also be given to the fact that, in contrast to the uncorrected curve, in the range of the crystallization the linear relation between the temperature and the time, which is characteristic for the DSC machine, is no longer valid. The internal sample temperature a t the peak maximum has an almost stagnant value, as in a thermogram characteristic for the solidification of metals. Knowing the crystallization temperature, we made a cross section of the sample taken from the pan of the DSC machine and counted the number of spherulites per unit surface in the middle of the cross section. Taking the 1 3 t h power of this number, one gets a fair estimate of the number of spherulites per unit of volume. The results of this procedure are shown in Figure 3. This is the only relation between the number of spherulites per unit volume and the crystallization temperature for isotactic polypropylene (i-PPI ever obtained. One may notice that the temperature dependence is quite considerable. If the total crystallization would occur exactly at the (corrected)peak temperature, the number of spherulites should exactly be equal t o the number of nuclei N(T) at this temperature, on
condition that our model, as explained at the end of the preceding section, is realistic. In principle, the shape of the DSC curve should give additional information on the crystallization kinetics. Unfortunately it turns out that the accuracy of the curve is insufficient for that purpose. Another result of Figure 3 is that the temperature range is restricted at the lowtemperature end. This is due to the fact that, at larger cooling rates, the control system of the DSC machine fails before the temperature range of the peak is reached during cooling (Janeschitz-Kriegl (1994)). The reader may recognize the rather complicated features of this type of thermal measurements. For larger cooling speeds, which are characteristic for the mold-filling process in injection molding, an apparatus is being constructed in our laboratory, by which we hope to imitate the characteristic features of heat transfer in this process, but avoid complications by flow. (Flow has been treated separately, as already indicated in the Introduction.)
The Growth Rate as a Function of Temperature At low degrees of supercooling the growth rate of spherulites of i-PP can be measured in the thermomicroscope. This has been done previously by von Falkai (1960) and Padden and Keith (1959). Also the so-called zone crystallization, where a rod of the material confined in some tubing is moved out of a stove in the direction of a negative temperature gradient, has been used for this purpose by Lovinger et al. (1977). For the realization of lower crystallization temperatures, however, these methods are not useful. In our laboratory several alternatives were tried. Of these alternatives two will be quoted in this review. A third one, relying on light scattering, will be described elsewhere. In one of our successful measurements the progress of a transcrystallization front against a steep temperature gradient was used by Ratajski (1993). The latter gradient was introduced in order to suppress the diffise zone crystallization (see Figure l), which hampers transcrystallization. For this purpose a rather thin tablet of the polymer was prepared and melted between confining metal plates. At time zero only one of these walls was quenched to the desired crystallization temperature. For such a situation it is known that-ignoring the changes in the heat diffisivity with temperature and crystallinity-a linear temperature profile between the two walls is established a t a Fourier number Fo > 0.5. This means that at observation times
t
>
0.5(D2/a)
(7)
the growth of the layer can be observed in a practically constant temperature gradient. (For D = 1 mm and a = lo-' m2/s one has t > 5 s.) If for the vicinity of the temperature of the quenched wall an exponential dependency of G(T) on temperature is assumed according t o the equation
G(T>= G, expl-B(T - T,)I
(8)
where G, is the growth speed a t the wall temperature T,, one can solve the differential equation governing the advance of the front analytically. By a comparison of experimental and theoretical curves G , and /?can be obtained. This procedure gives much more acccurate values of G, than the visual determination of the initial slope of the experimental curve which, in principle,
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3486
-5
*
. . ,
. :. . . . . . . . . . . .
.
:.
g
. . . . .. . . . . . . . .
:
~
p-polypropflene radial growth rate
.
-5
I".. :
.^
; . .......... 1.. ........
g ....
~
. . . . .
:*
4
-H a-Polypropy(ene radial growth rate
...........
...
.u
-
.I...
...
4
, 90
*
Ratsil*U~tiul Ratslskllmnlgm*m dats
1
LovmWT')
.
, IW
.
~
&
.*
. . . . . . :. . . . . . . . . . . . . .
, 110
.
RatsWhicmtawt.
L .
,
im
.
, 130
.
,
1. i
140
Temperature ("C)
Figure 4. Growth speed of &spherulites of i-PP as a function of temperature, as obtained from modified transcrystallization experiments (filled circles), from optical microscopy (filled squares), and from zone crystallization by hvinger et al. (1977) (open triangles).
:
LOUinpSP~
*
'-I . a
-10
.
.
.... :
P.ddenrKaimm wn Falkai"l
a w e ' :
; . ; . ; . , . ;
M
"
80
1W
110
.
. , . , . ; . ,
120
IM
140
150
-10
160
Temperalure ('C)
Figure 6. Comparison of the results of the modified transcrystallization (filled squares) and of the thin layer (filled circles) methods with respect to the growth speed afa-spherulites ofi-PP as a function of temperature. Various literature data concerning the high-temperature range are indicated by open symbols, showing that this growth speed is independent ofmolecular mass.
obtained directly with this method. On the other hand, one can easily calculate the growth speed of the a-modification from that of the @-modificationwith the aid of the observed cone angle ("aperture" K ) , as pointed out by Lovinger et al. (1977). In fact, one has
-GQ _ - cos zK
(9)
GP
Figure 5. Transcrystallized layer of i-PP (modified method), after a quench on the lower side from 240 to 110 "C and a contact time of48 s. Notice the bright / l a n e s dominating the growth surface. The a-growth is represented by the striated field. Only the apex of one ofthe cones is in the surface ofthe cross section.
should furnish G,, but is inaccurate because of the scatter of the experimental points. The described method was successfully applied to the determination of the growth speed of the @-modification of i-PP. For the purpose, the surface of the quenched wall was rubbed with a nucleation agent for this modification (Cinquasia-Gold). The results of this investigation are shown in Figure 4. As one may notice from this figure, the temperature range, which was limited in the classical measurements of Lovinger et al. (1977)(open triangles) by a lowest temperature of -120 "C, could be extended down to 90 "C. The question may now arise of why this investigation into a property of the @-modification is of such a n advantage, where with normal bulk crystallization the a-modification is predominant. The answer to this question is given by Figure 5. This figure shows a cross section through a sample showing a transcrystallized layer. This sample was obtained with a n untreated wall. In spite of the fact that, apparently, only a few nuclei of the @-modificationwere present a t the wall, the faster P-growth from these nuclei, recognizable as bright cones between crossed polars, starts to dominate the layer growth after a short time. This means that the lower growth speed of the a-modification cannot be
The values of Ga(T),as obtained in this way from the data presented in Figure 4, are given in Figure 6 by filled squares. This figure also contains a great number of data previously obtained by other authors in the hightemperature range (open symbols). Also here the extension into the range of lower temperatures is quite remarkable. However, because of the indirect way of the described determination, it seemed desirable to have an independent check. For this purpose crystallization was observed in thin layers. As is well-known, in a thin layer a particular spherulite can laterally grow out to a much bigger size, before growth is stopped by impingement with other spherulites, as was shown by Chew et al. (1989) and Billon et al. (1989). The reason for this fact is that growth from nuclei in neighboring layers, which occurs in the bulk, cannot interfere. A condition is that the considered layer is embedded between walls which themselves are inactive with respect to nucleation. The surfaces of thin cover glasses appeared quite appropriate in this respect. Thin layers were produced in a microtome. The obtained slices of about 5 pm were melted between cover glasses. The obtained sandwiches were quenched for the desired crystallization time in a bath of diethylene glycol of properly chosen temperature and, afterward, quenched finally in tap water. If for every bath temperature the radii of the largest spherulites were plotted against the crystallization times, neat straight lines could be drawn through the obtained points. For every bath temperature growth speeds could be derived from the slopes ofthese lines. These growth speeds are inserted in Figure 6 as closed circles. One observes that the positions of these points are a little lower than those of the points obtained according to the previous method. Nevertheless, this fair agreement
3486 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
shows that both methods are realistic. In fact, the points obtained from thin slices can lie a little lower, if the growth at the rims of the virtually two-dimensional spherulites (disks) is retarded a little bit by the glass surfaces. The usefulness of Figure 6 is also given by the fact that it contains the maximum growth speed of a-spherulites. Unfortunately, on the low-temperature side no data are available so far for i-PP. However, according to Magill et al. (1973) for slowly crystallizing polymers the complete curve should practically be symmetric with respect t o the temperature at the maximum. On the logarithmic scale chosen it should also go to --m a t the glass temperature of -4 "C. This, of course, holds for the (questionable) assumption that crystallization follows the same pattern at lower temperatures. In a next paper it will be shown that the difficulties are even more serious for high-density polyethylene.
-
Conclusions Emphasis is placed in this paper on the fact that the structure of the final product is determined by the interplay of nucleation and growth, as occurring in the course of the heat transfer process, where not only the nucleation in the bulk but also that a t the confining mold surfaces can be of importance. The proper mathematical tools can be found in previous papers, viz., the set of rate equations, as given by Schneider et al. (1988) for the bulk, as well as the extended equations including the wall influences, as formulated by Eder (1995). It is also mentioned that these equations are differential equations by nature and thus can be integrated together with the equation of heat conduction with boundary conditions. These equations are coupled by the heat of crystallization gradually evolved during the process of crystallization in a medium of varying degrees of supercooling. As in principle the spherulitic growth depends not only on the local temperature, but also on the local temperature gradient, an upper bound has been given for the admittable gradient in terms of the local number of nuclei. In the present paper the difficulties which are encountered with the collection of the necessary kinetic data, even if the influence of flow is excluded, are summarized. These difficulties are particularly large for those commodity polymers which crystallize a t a reasonable speed. It is shown that probably the best approximation is the use of a number of nuclei N which is a unique function N(T) of temperature. Such a function corresponds t o an activation time spectrum in which every activation time jumps from infinity to zero when, during cooling, the specific activation temperature of the envisaged nucleation process is reached. In this connection, the demonstration of the combined thermal and optical method, as used for the determination of N(T) of i-PP, seems very instructive. In the corresponding Figure 3 for the first time a realistic quantitative picture of the increase of the nucleation activity of i-PP with decreasing crystallization temperature is given (on the mentioned crude basis, of course). However, at the moment there is no better approach for a prediction of the influence of the thermal history on the structure of the product. Similar difficulties are encountered when the growth speeds of spherulites are to be determined at lower crystallization temperatures, which are a bit more characteristic for the real industrial crystallization processes. Fortunately, two independent methods led t o fair agreement (see Figure 6).
One point emerges clearly from these measurements: There must always be a method to get hold of the temperature at which certain structures are actually formed. It seems impossible to deduce crystallization kinetics as function of crystallization temperature if only the final structure of the end product can be investigated.
Acknowledgment The authors are indebted to Dr. Christian Paulik for assistance with some measurements (Figure 3) and to Mr. Manfred Lipp, who, as an instrument maker, never hesitates to dedicate his full capacities t o constructing and altering unusual measuring devices. The authors also express their thanks to Petrochemie Danubia for cosponsoring the work of one of us (E.R.) under Contract Nos. 51594, 51570, 51548, and 51522 of the Austrian Foundation for the Support of R & D (Forschungsforderungsfonds fur die gewerbliche Wirtschaft) and to the Austrian Foundation for Scientific Research (Fonds zur Forderung der wissenschaftlichen Forschung) for sponsoring a continuation of this work of E.R. under Contract No. P 10284.
Literature Cited Astarita, G.; Kenny, J. M. The Stefan and Deborah Numbers in Polymer Crystallization. Chem. Eng. Commun. 1987, 53, 69. Avrami, M. Kinetics of Phase Change. J. Chem. Phys. 1939, 7, 1103; 1940, 8, 212; 1941, 9, 177. Berger, J.; Schneider, W. A Zone Model of Rate Controlled Solidification. Plast. Rubber Process. Appl. 1986, 6, 127. Billon, N.; Esceleine, J. M.; Haudin, J. M. Isothermal Crystallization Kinetics in a Limited Volume. Colloid Polym. Sei. 1989, 267, 668. Carslaw, H. C.; Jaeger, J. G. Conduction of Heat in Solids, 2nd ed.; Clarendon Press: Oxford, 1959; p 282. Chew, S.; Griffith, J. R.; Stachurski, Z. H. The Crystallization Kinetics of Polyethylene under Isothermal and Non-Isothermal Conditions. Polymer 1989, 30,874. Eder, G. Fundamentals of Structure Formation in Crystallizing Polymers. Macromolecular Design of Polymeric Materials; Hatada, R,Kitayama, T., Vogl, O . , Eds.; Marcel Dekker: New York, 1995, in press. Eder, G.; Janeschitz-Kriegl, H. Heat Transfer and Flow: Transport Phenomena Controlling the Crystallization Process of Polymers. First Int. Conf. Pacific Inst. Thermal Eng.; Technomic Publ. Corp.: 1993; p 1031. Eder, G.; Janeschitz-Kriegl, H.; Liedauer, S. Crystallization Processes in Quiescent and Moving Polymer Melts under Heat Transfer Conditions. Prog. Polym. Sci. 1990, 15, 629. Falkai, B. von. Melt and Crystallization Behavior of Macromolecular Substances I. Investigation of Crystallization Kinetics of Isotactic Polypropylene (in German). Makromol. Chem. 1960, 41,86. Janeschitz-Kriegl, H. Fundamentals of Texturing: Crystallization by Heat Transfer and Flow. Macromol. Symp. 1994, 83, 207. Janeschitz-Kriegl, H.; Wippel, H.; Paulik, Ch.; Eder, G. Polymer Crystallization Dynamics, as Reflected by Differential Scanning Calorimetry I. On the Calibration of the Apparatus. Colloid Polym. Sei. 1993,271, 1107. Kolmogoroff,A. N. On the Statistical Theory of the Crystallization of Metals (in Russian). Izv. Akad. Nauk SSSR,Ser. Math. 1937, 1, 355. Liedauer, S.; Eder, G.; Janeschitz-Kriegl, H.; Jerschow, P.; Geymayer, W.; Ingolic, E. On the Kinetics of Shear Induced Crystallization of Polypropylene. Int. Polym. Process. 1993, 8, 236. Liedauer, S.; Eder, G.; Janeschitz-Kriegl, H. On the Limitations of Shear Induced Crystallization in Polypropylene Melts. Int. Polym. Process. 1995, in press. Lovinger, A. J.; Chua, J. 0.;Gryte, C. C. Studies of the a- and /%Form of Isotactic Polypropylene by Crystallization in a Temperature Gradient. J. Polym. Sei., Phys. Ed. 1977,15,641.
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3487 Magill, J. H.; Li, H. M.; Gandica, A. A. Corresponding States Equation for Crystallization Kinetics. J. Cryst. Growth 1973, 19, 361. Malkin, A. Ya.; Beghishev, V. P.; Keapin, I. A.; Bolgov, S. A. General Treatment of Polymer Crystallization Kinetics. Polym. Eng. Sci. 1984,24,1396. Mandelkern, L.Crystallization of Polymers; McGraw Hill: New York, 1964. Nakamura, K.; Watanabe, T.; Katayama, K.; Amano, T. Some Aspects of Nonisothermal Crystallization of Polymers. J.Appl. Polym. Sei. 1972,16, 1077. Olley, R. H.; Bassett, D. C. On the Development of Polypropylene Spherulites. Polymer 1989,30,399. Padden, P. J., Jr.; Keith, H. D. Spherulitic Crystallization in Polypropylene. J. Appl. Phys. 1969,30,1479. Ratajski, E. Crystallization Phenomena in Quiescent Polymer Melts (in German). Doctoral Thesis, Linz University, March 1993. Schneider, W.; Koppl, A.; Berger, J.; Non-Isothermal Crystallization of Polymers. Znt. Polym. Process. 1988,2,151.
Schulze, G. E. W.; Naujek, T. R. A Growing 2D Spherulite and Calculus of Variations. Colloid Polym. Sci. 1991,267,689 and 695. Stefan, J. On the Theory of Formation of Ice, in Particular in the Polar Sea (in German). Ann. Phys. Chem. (Wiedemann) 1891, 42, 269. Wu, Ch.; Eder, G.; Janeschitz-Kriegl, H. Polymer Crystallization Dynamics, as Reflected by Differential Scanning Calorimetry 11.Numerical Simulations. Colloid Polym. Sei. 1993,271,1116. Wunderlich, B. Macromolecular Physics; Academic Press: New York, 1973;Vol. 1, pp 282,336. Received for review January 9,1995 Revised manuscript received June 5 , 1995 Accepted J u n e 20, 1995@
IE9500331 Abstract published in Advance ACS Abstracts, August 15, 1995. @
