# Industrial solidification processes in polybutene-1. Part I-quiescent melts.

INTRODUCTION. SOLIDIFICATION IN A QUIESCENT SAMPLE

For the description of the process of solidification of a crystallizable sample, several theories have been proposed. The first classical theory on this subject was published by Stefan in 1891 using a method developed by Neumann in 1868 (see Carslaw and Jaeger (1)). It is based on the simple assumption that the temperature at which crystallization takes place is invariable and equal to the equilibrium melting point. A melt is considered, which is at some uniform temperature T above the melting point [T.sub.m] and fills the half infinite space, x > 0. At time t = 0 the wall at x = 0 is suddenly quenched to a temperature well below the melting point. This initiates the transition into the solid state. It is also assumed that the thermal properties as specific heat, heat conductivity and density would be independent of the temperature but might be different in both phases. This leads to a simple solution of a heat transfer problem with a free boundary between liquid and solid state, moving into the melt. In this way one obtains the famous square-root law:

[x.sub.c](t) = a[square root of (t)] (1)

which means that the thickness of the solidified region [x.sub.c](t) increases with the square root of time after the sudden quench. At short times a fast process with infinite initial speed is predicted in this way, whereas the assumption made in this connection holds only for processes that proceed slowly enough (quasi equilibrium). This means that this theory is inconsistent.

A thorough reconsideration of this problem, which is the simplest heat transfer problem with phase transition, leads to a modified front theory. Beginning with the assumption that crystallization starts from nuclei located at the wall surface one can imagine that, if the nearest neighbor distance between those nuclei is small enough, a crystallization front is formed near the wall. According to a theory by Eder and Janeschitz-Kriegl (2), the initial speed of progress of this front is finite and equal to the growth speed of spherulites at the temperature of the quenched wall. But the main difference to the classical Stefan theory is the fact that, together with the supercooled crystallization front, also some zone ahead of the crystallized layer is supercooled (3) in the melt. So nucleation in the said zone of the fluid can no longer be excluded, as it has been in the classical theory, where the melt is always above the thermodynamic melting point and thus free of nuclei.

In a further development of The Stefan theory it is shown that, due to diffuse nucleation in the bulk of the fluid, a diffuse crystallization zone comes into existence (Berger and Schneider (4)), which starts with some delay but moves at a higher speed into the melt than the said front, superseding that front earlier or later. With raising of the overall speed of crystallization, for example by the addition of some nucleating agent, the zone becomes narrower and supersedes the front at an earlier time. If the crystallization speed is infinite, one obtains the classical front movement. Furthermore, the location of the zone is almost independent of the number of nuclei at the wall. If this number is zero, no crystallization front can be observed in the beginning. This type of front-like growth has been called transcrystallization (Wunderlich (5)). Figure 1 is a schematic drawing representing these theoretical results. Actually. Fig. 1 holds for an infinite Biot number. This means that the heat conductivity in t he wall must be extremely high compared with that in the (polymer) melt (6), which, in particular, is never the case with metal casting (7).

A statistical treatment of crystallization processes was first given by Kolmogoroff (8). However. Avrami's work (9), though more special and developed later, became more famous. If the kinetic data (nucleation rate and growth speed of spherulites) are known as functions of temperature, one can calculate the mentioned structure formation for a given temperature history, which, however is never uniform with practical cooling processes. As a consequence, more powerful means are required. Figure 1 should convey some feeling for these problems (see later).

Avrami's theory describes the time dependence of crystallization and was originally developed for the crystallization of metals. This theory leads to the introduction of two different volume fractions:

a) the "unrestricted volume fraction" [[phi].sub.0](t)), which describes the transformed volume per unit of volume, if the spherulites do not collide with each other, and density changes are ignored.

b) the "covered volume fraction" [xi](t), which occurs, if impingement of the spherulites is taken into account, density changes still being ignored.

According to Avrami the covered volume fraction can be calculated as:

[xi](t) = 1 - exp(-[[phi].sub.0](t)). (2)

where [xi] is the volume fraction actually covered by spherulites and [[phi].sub.0](t) is the unrestricted volume fraction, which is a function of the growth speed G and of the nucleation rate [alpha]. More explicitly one has:

[xi](t) = 1 - exp(-4/3 [pi] [[integral].sup.t sub.0] [alpha](s) ds[[[[integral].sup.t sub.s] G(u) du].sup.3]). (3)

where G(u) is the time dependent growth rate and [alpha](s) is the time dependent nucleation rate. Mostly, these time dependencies are indirect, as a consequence of the changing temperature (see Eq 4 below).

In order to describe real heat transfer processes, in which the temperature history is not uniform, Schneider et al. derived a system of differential equations (10) ("rate equations") by successively differentiating Kolmogoroff's Eq 3 with respect to time (see also Eder Et al. (3.11)).

The local degrees of crystallinity, as well as the local structures (numbers of spherulites per unit volume or mass) can then be calculated correctly by solving the said system of differential equations together with the equation of heat conduction, which is also a differential equation. For this purpose one is permitted to assume a number of (athermal) nuclei per unit volume, N(T), as a unique function of the crystallization temperature. This has been shown elsewhere (11-13). In this respect the only condition is that the crystallization temperature T is not too close to the thermodynamic melting point, where sporadic "thermal" nucleation can be expected. Under the conditions, as just specified, one simply has:

[alpha] = dN/dT dT/dt (4)

In quiescent melts this package of equations permits the treatment of complex heat transfer problems accompanying solidification. For a pertinent application on DSC the work by Wu et al. (14) should be mentioned.

Fortunately, the said temperature range close to the thermodynamic melting point is quite irrelevant in quiescent polymer melts. In fact growth speeds G are extremely low in this upper range of temperatures (see e.g. Fig. 2). Because of the occurrence of the third power of G in the kinetic equations (see Eq 3) crystallization kinetics is extremely sluggish in this temperature range (with probably HDPE as an exception). But this means that, as a consequence of the usually high cooling rates of industrial processes, structure development only occurs at low temperatures, where athermal nuclei are dominant.

The just mentioned influence of the cooling rate can be illustrated by an equation given by Eder (15). This equation describes the critical cooling rate [q.sub.crit] = [(dT/dt).sub.crit], which must be applied in order to transfer the melt into the glassy state (1% crystallinity as a measure). In such a fast process crystallization does not get enough time for its development. One has:

[q.sub.crit] = 13.24 [G.sub.max] [N.sup.1/3.sub.max]/[K.sub.G], (5 )

where [G.sub.max] is the maximum cooling rate of the bell-shaped curve G(T) with zero values at [T.sub.m] and [T.sub.g], [(1/[K.sub.G]) is its width, and [N.sub.max] is the number of nuclei per unit volume at the temperature [T.sub.max] where the maximum occurs. By the way, such a simple equation is obtained only because heat transfer is not explicitly treated. The (uniform) cooling speed is just assumed. (In principle, nearly uniform cooling can be achieved only with extremely thin samples and, preferably, in the absence of the generation of latent heat, as is the case with Eq 5.) So, the mentioned set of "rate equations" was not needed in this special case. In an early stage of the development Gandica and Magill (16) published a "universal" shape of G(T), as obtained on rather slowly crystallizing polymers. The pertinent measurements on fast crystallizing industrial polymers required quite new techniques, which are presented amongst others in the present publication. Also the determination of the number N(T) has to be developed for these polymers. At the end of this paper the usefulness of Eq 5 for an illustration of the situation with PB-1 will be demonstrated.

At the end of this section it should be mentioned that, unfortunately, some theoretical expressions preceding the mentioned "rate equations" (3, 10, 11) are still in vogue (see e.g. K. Nakamura et al. (17)). These latter authors relate the overall crystallization speed to the momentary degree of crystallinity. However, in this way the influence the temperature history on the morphology is wiped out (11). In fact, everyone knows that, with increasing cooling speed, the morphology becomes more fine grained. Only the said "rate equations" permit a correct treatment: the speed of crystallization is proportional to the momentary crystallite surface, which is a much more cumbersome condition from the viewpoint of mathematics.

CHARACTERIZATION OF POLYBUTENE-1

Isotactic polybutene- 1 (simply PB-1) with a basic formula --[[[CH.sub.2]-CH([C.sub.2][H.sub.5])].sub.[n.sup.-]] was first prepared by Natta et al. in 1955 (18). This polymer is manufactured in relatively small tonnages for special purposes only. and therefore is less familiar than either polyethylene or polypropylene.

Usually polybutene- 1 is produced either as a homopolymer or as a (random) copolymer with an ethylene content in the range from 0.75% to 5.5%. Because of the slow decline of its tensile modulus with increasing temperature and its superior creep behavior PB-1 is mainly used in hot water pressure pipe applications. Another application is as an additive in FE for the manufacturing of easy opening packages

A feature that does not facilitate a more extensive use of PB-1 is its polymorphism. When crystallizing from the melt it first adopts "Form II" as a crystalline modffication. This modification is unstable and gradually transforms into Form I. This process results in changing properties.

For the present work, polybutene- 1 grades of Melt Flow Rates (190/2.16) of 0.4 and 4, and a density of 0.915 g/[cm.sup.3] (ASTM D 1505 samples conditioned for 10 days at 23[degrees]C), were used. Their code names at the Shell Chemical Company are PB0110 and PB0300. At 200[degrees]C the zero shear viscosity of PB0110 is about fifty times that of PB0300. PB0110 is nucleated with a small amount of HDPE. PB0300 does not contain a nucleation agent. Both polymers are heat stabilized.

Currently these products are marketed through Basell Polyolefins. The melting temperatures of PB-1 are reported at 112[degrees]C (Form II) and 124[degrees]C (Form I) (see data sheets of Basell Polyolefins).

Previously, the melting temperatures of PB-1 were reported at 124[degrees]C (Form II) and 136[degrees]C (Form I)(19, 20) and the glass transition temperature at -24[degrees]C (21). Aycock and Wunderlich (22) report a molar heat capacity around 110 J/molK (at 100[degrees]C). which is about twice the heat capacity of polypropylene and four times that of polyethylene. Apparently. the comonomer content in the Basell products causes the lower melting temperatures.

The mentioned studies indicated the existence of the two modifications: a stable twinned hexagonal form, called "Form I," where the polymer chains are arranged in a 3/1 helical conformation and an unstable tetragonal form. "Form II" with the chains assuming an 11/3 helical conformation. If crystallized from the melt, PB-1 first assumes the tetragonal form. which, depending on factors such as storage temperature, mean molecular weight, molecular weight distribution, comonomer content, gradually undergoes a transition into the hexagonal form on storage or under the influence of stress. A third modification of an untwinned hexagonal form could be obtained by precipitation from solution (20) or from the melt under high pressure (23).

The Form II [right arrow] Form I phase transformation, which is accompanied by an extension of the helical axis from 0.187 nm to 0.218 nm per chemical repeating unit (24), results in an increase of crystalline density and significantly alters the properties of the material. The reported crystalline melting temperature increases from 124[degrees]C to 136[degrees]C and the material becomes increasingly turbid, more rigid and exhibits higher strength.

A lot of work has been done to characterize the nature of this transformation. For example, Boor and Mitchell (19) examined the effects by means of dilatometric contraction measurements in a density gradient column and by infrared spectroscopy. They found a pronounced maximum in the rate of conversion of Form II to Form I between 15[degrees]C and 20[degrees]C, which has been confirmed in other studies. Powers, Hoffmann, Weeks and Quinn (25) analyzed the change of the degree of crystallinity in this process (51.8% in Form II [right arrow] 77.1% in Form I. The ex-reactor PB0110 presently has degrees of crystallinity changing from 45% to 55%-60%). The just-mentioned authors postulated that the transformation is a nucleation-controlled, interfacial transport-process, and that the change of the helix conformation occurs in a given lamella without a reorganization of this lamella with respect to others. This fact would mean that numbers of spherulites (i.e. nuclei) per unit volume should not change by this transfor mation. Growth speeds should be those of Form II, of course. In fact those speeds would govern the heat transfer. Information concerning the molecular mechanism of transformation was obtained by several authors (5, 26) who carried out Avrami-type analyses. Receiving an Avrami-Index of n = 2, they concluded that the nucleation of the stable form is instantaneous.

GROWTH SPEED OF SPHERULITIES

For the purpose a thermo-microscope (Nikon optihot-2 Linkam) was used for the higher temperatures and Thin-Slice-Experiment II, as recently described by Stadlbauer et al. (27), for the lower temperatures. Thin slices have been used (11, 28, 29) for the purpose of drastically reducing the number of spherulites per unit field of vision in the microscope.

For the experiments on the thermo-microscope slices with a thickness of 5 [mu]m were cut with a micro-tome directly from the granules and put between two glass slides on the hot stage of the microscope. They were melted and kept for 5 minutes at 200[degrees]C and subsequently cooled to the intended crystallization temperature at a cooling rate of 130[degrees]C/min. As soon as the final temperature was reached photographs were taken at certain time intervals. The radii of the growing spherulites were then measured and plotted against time. The slopes of the linear least square plots were taken as the radial growth speeds G.

For the thin-slice experiments, a drop of a 3% solution in p-xylole was placed on a cover glass. When this drop dried, it formed an extremely thin PB-1-film, of the order of 0.1 [mu]m, which was covered by a second cover glass. In this apparatus extraordinarily fast quenches could be carried out from a temperature of 180[degrees]C to the crystallization temperature, and from there to a lower temperature, where the momentary structure was immobilized. In this way, a very large number of extremely small spherulites filled the space between the much bigger spherulites, which grew at the crystallization temperature. In this apparatus the times at the crystallization temperature could be defined nicely, so that a plot of the spherulite radii vs time could easily be constructed. The slopes of the linear least square plots were taken as the radial growth speed G (see Figs. 4a and 4b).

The results of both experiments are shown in Fig. 2 for PB0110 and PB0300, respectively. Obviously, the maximum of the growth speeds lies a little lower than 70[degrees]C for both polymers. This maximum seems a little higher for PB0110. For both curves quadratic approximations were constructed, which are given in Eqs 6a and 6b.

For PB0110 one gets the quadratic approximation:

log G(T) = - 5.82 - 0.00213[(T - 74.88).sup.2] (6a)

and for PB0300:

log G(T) = - 5.95 - 0.00147[(T- 66.33).sup.2] (6b)

It remains questionable whether the growth speeds of these two samples differ significantly. The uncertainties on the low temperature side are of minor importance for a simulation of the solidification process.

NUMBERS OF NUCLEI

PB-1-granules were melted and pressed in vacuum to a compact cylinder of a diameter of 20 mm and a length of 200 mm. Out of this cylinder several smaller rods with 20 mm length and 4 mm diameter were machined, An apparatus, which was developed at the University of Linz, was used (13). This apparatus applies the counter-current principle: a cold stream vs. a warm stream. It enables one to melt a rod (produced as described above) in a gaseous environment by radiation, quench it first with a very high cooling rate (application of the cold stream) and arrest it at a certain nucleation temperature (by the warm stream).

For this purpose the sample was wrapped into an aluminum foil, and suspended in the coil of a thin wire. A thermocouple, which is placed in an axial hole in the center of the sample, is connected to a computer, which records the temperature-time-profile. In our experiments the sample was heated up to 180[degrees]C and held there for app. 10 mm. It was then quenched down to a temperature close to the intended crystallization temperature with cold water. in order to arrest it at this temperature it was rinsed with ethylene-glycole of the proper temperature. After 20 minutes the sample was slowly cooled to room temperature and removed from the apparatus. Thin slices were cut with a microtome normal to the axis of the rod.

Slices with a thickness of 5 [mu]m were embedded in Canada-balsam on object holders and covered with thin cover glasses. The number [N.sub.N,A] of spherulites per unit area of the cross section was' then counted on a photograph, as taken under a microscope. These numbers were divided by the area covered by the photograph and raised to the 1.5th power for rendering the number of spherulites per unit volume:

[N.sub.N,V] = [([N.sub.N,A]).sup.3/2] (7)

(As has recently been shown by Stadlbauer et al. (27), this equation furnishes a very acceptable approximation of the real number per unit volume. The results obtained are shown in Fig. 3. The acceptability of this approach was shown explicitly for two polyketones. In fact, at suitable crystallization temperatures, where the number of nuclei was not too large, this number could directly be counted in a (tiny) volume defined by the said area under the microscope and the (very low) thickness of the sample (Stadlbauer et at. (27)). The number of nuclei counted in this way was a little lower than the numbers obtained with the method described in the present paper. The reason for this fact may be found in a certain under-shoot (of the order of 10[degrees]C), as observed with the counter-current method. Nevertheless, the counter-current method must be preferred, because of the much larger range of accessible crystallization temperatures and the much easier handling.

For PB0110 being nucleated, the number of nuclei per unit volume is much higher, than for PB0300 in the whole range of temperatures. The points in the high temperature range (near 110[degrees]C) are less reliable, as in this range sporadic nucleation plays a role. In Fig. 3 it can be observed that, with decreasing temperature (below 80[degrees]C), the rows of points for both samples converge. But this means that, at low enough temperatures, the nucleation agent loses its influence. At those temperatures only the athermal nuclei, which act in the unnucleated sample, seem to retain their influence.

DISCUSSION

At temperatures near 110[degrees]C sporadic nucleation is reflected in a convincing way by a plot of the radius of a selected spherulite vs time, as obtained with the microscope (see Figure 4a). In fact, a negative intercept with the ordinate axis is found in such a case. This indicates a belated birth of the pertinent nucleus. In fact, under the microscope the growth of an individual nucleus can directly be followed only at such relatively high temperatures. In contrast, a collective of nuclei is always considered in the thin slice experiments. The success of the thin slice experiment is based on the fact that, at lower temperatures, the size of all growing spherulites is the same. Apparently they all start growing simultaneously. For comparison a plot is shown in Fig. 4b, as obtained at a temperature close to the temperature of the growth maximum in Fig. 2. The straight regression line practically meets the origin without constraint. Look at the very different time scales on the abscissae of Figs. 4a and 4b! At temperatures below 70[degrees]C, however, positive intercepts with the ordinate axis are regularly found, in spite of the very fast quenches realized in the thin slice experiments. It appears that crystallization near 70[degrees]C is so fast that spherulites start growing. when the sample passes this temperature before the chosen final crystallization temperature is reached. However, because of the apparently very high reproducibility of the quenches, this positive intercept remains the same for all samples quenched to the same crystallization temperature and kept there for various times. As a consequence, one still obtains a straight line with--admittedly--a certain intercept at the ordinate axis. From the slope of this line one can readily calculate the growth speed at those temperatures.

If Eq 5 is used for a calculation of the critical cooling speed with the aid of the data of Figs. 2 and 3, one obtains values in the neighborhood of 25K/s dependent to some extent on whether the nucleated or the un-nucleated sample is taken. Apparently, such a high critical cooling speed cannot be reached with our quenching technique with, as a consequence, the mentioned positive intercept at the ordinate axis.

For other polymers interesting values for [q.sub.crit] have been found: 6000 K/s for HPPE, 600 K/s for a polyketone, 60 K/s for i-PP and 1 - 10 K/s for PET, dependent on the copolymer content. For a judgment of the usefulness of DSC measurements these values have to be multiplied by 60, as cooling rates in DSC are usually given in K/min. In fact, 100 K/min is a rather high value in polymer research. This means that the usefulness of DSC measurements is questionable for the present purpose. For PB-1, one would need 1500 K/min and for i-PP 3600 K/min, not to speak of HDPE.

CONCLUSIONS

In the experiments described data were obtained, which enable the calculation of heat transfer and structure formation in quiescent melts of the investigated polymer samples. This paper may serve also as an introduction to the second paper of this series concerning the shear induced crystallization of PB0110.

Unfortunately, an interpretation of the data on a molecular level has not been tried so far. Such an interpretation seems extremely difficult because of the cooperative nature of the mechanisms involved. It goes without saying that the lack of data for the relevant temperature ranges has been impeding such an endeavor in the past.

[FIGURE 1 OMITTED]

[FIGURE 2a OMITTED]

[FIGURE 2b OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4a OMITTED]

[FIGURE 4b OMITTED]

ACKNOWLEDGMENTS

This work was carried out under the auspices of the Brite-EuRam project "Structure Development during Solidification in the Processing of Crystalline Polymers" (DECRYPO). The Shell-Company Louvain-Lap Neuve is acknowledged for permitting the publication of the results obtained on their industrial grades of polybutene-1.

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