Graph 1
I can see some heads nodding: of course we understand crystallization from the melt in polymers, why ask?
Really? Not so fast!
In the Graph above we report some interesting observations done with a differential scanning calorimeter (DSC) on the cooling of a PET melt, measuring its crystallinity from the crystallization peak, followed by heating to melt the crystals and obtain a melt again: a classic! The basic idea is that what has crystallized, one way or another, must melt! This should be really easy to verify from the DSC traces on cooling and on heating by integration of the crystallization and melting peaks, after subtraction of the baselines.
In this Graph the x-axis is the %X calculated from the crystallization peak on cooling, in other words the amount of crystallinity obtained under the cooling rate (10 oC/min). This should be the final crystallinity in the sample at room temperature when cooling is done. It should be the crystallinity value that we find from the heating curve after we re-heat the sample up to recreate a melt.
For PET, which does not crystallize fast, unlike PE or PP, crystallization is far from complete and we can only obtain a maximum of about 55% crystallinity by cooling and subsequent annealing to increase the crystallinity level. If we cool the melt very fast, crystallization hardly occurs on cooling but we observe a “cold-crystallization” peak on re-heating at about 60 oC above the Tg. Of course then, on re-heating, we need to account for the amount of cold-crystallization when calculating the degree of crystallinity the sample had at the beginning, which we call Xo% in the Graph. When no cold-crystallization peak is observed on heating, we simply integrate the melting peak to determine %Xo; otherwise the correction to account for the added crystallinity due to cold-crystallization is straightforward.
Now let us compare the value of Xo% plotted as the y-axis on this graph versus X% obtained from the cooling curve. The red line Y=X is where we are expecting to find all the points!
What are all these various points representing?
PET is sold by the resin manufacturers as small white pellets which one needs to dry extensively before exposing them to any thermal treatment, such as in a DSC or to measure their melt index MFI or to extrude. In an extruder, the pellets are melted by heat to approximately 300 oC, then the melt is sheared, goes through a die to form a parison for instance, or a strand, a film, a sheet, fibers etc. The part is then cooled rapidly and we can cut out a piece of it, dry it again, and study it in a DSC. In the experiments I performed to obtain the data points in Graph 1 above, I inserted a Rheo-Fluidizer flange in between the end of the extruder and the die, so I could create a thermo-mechanical history for the melt before it goes to the die to form the part and cool. The various points correspond to DSC traces of samples obtained from melts with various thermo-mechanical histories. The extruder rate was changed, the temperature in the Rheo-Fluidizer was changed, the gap dimension was changed, the shear rate and the vibration of the melt in the Rheo-Fluidizer were changed; and the post-treatment could also be a variable: some parts were annealed after they were formed, some of them were not. The annealing temperature and time were also varied. In other words the samples were all processed differently.
The objective of the study was the influence of the instability of the entanglement state, triggered by applying non-linear rheological conditions in the Rheo-Fluidizer, on the crystallinity. This is really a trivial series of experiments, easy to perform and easy to analyze.
But the result is not trivial at all and, in my opinion, implies far reaching theoretical conclusions about melt equilibrium and about crystallization from the melt!
Only 2 points on the Graph are on the y=x red line. Most of the points are below the red line, only 5 points are above it. Additionally, there appears to be a trend of two sets of points: one set gathering low %Xo data between 5 and 7.5%, and another set just below the red line with %Xo between 25 and 27.5%.
What does this mean?
The crystallinity calculated from the heating trace is below, sometimes much below (in one instance by 40%) the crystallinity acquired on cooling. Where did the crystals nucleated and grown during cooling go? Did they melt without leaving any thermal trace?
Obviously the method used to calculate the % crystallinity must be modified and some of our assumptions are wrong.
For instance, we have assumed that the exothermal peaks on cooling and heating was entirely due to the formation of the crystals. We have also assumed that the endothermal activity starting at the onset of melting up to the endset of melting was entirely due to the melting of the crystals. The classical pictures of a melt and of crystallization from it teach that, but are these assumptions correct?
In the dual-phase model of polymer interactions[i], the b-conformers form b-grains which can grow, melt, and multiply as if they were an amorphous form of condensate, actually more favorable to form than crystal nuclei. The formation of b-grains is exothermic and their melting is endothermic, like for crystals (but not with the same entropy nor enthalpy), so they should be included in the description of the DSC trace. In essence and to simplify, the b-grains formation and melting compete with the formation of crystals. Like for crystals, the b-grain population might be out of equilibrium due to kinetics reasons and annealing will try to make it return to equilibrium. All of this can be quantified: it’s called the Dual-Phase model of polymer interactions. When entanglements exist, at higher M (M>Me), the situation gets a little bit more complex, but not really when doing thermal analysis such as in a DSC. For entangled polymers we deal with “Crossed Dual-Phases” and the big difference with un-entangled polymers arises during a mechanical deformation of the melt: entangled melts are more sensitive to thermo-mechanical histories such as flowing in an extruder or through a die.
Yet there are still more troubling thoughts emerging from Graph 1 above: the %X on the x-axis is measured from the cooling curve of a melt starting at 300 oC. This melt is the result of heating the little piece of sample in the DSC pan from -25 oC to 300 oC, followed by an isothermal soaking period of 2 min at this temperature. I said “melt at 300 oC”. The melt state actually starts at the end of the melting peak around 250 oC, so at 10 oC/min heating ramp the melt is relaxing for 5 min to reach 300 and stays another 2 min at 300. According to classical views any thermo-mechanical history would be erased at 300 oC when the longest relaxation time at that temperature is of the order of a millisecond. If the melt has been able to relax for 5 min (300,000 ms) why do we see such differences for the crystallization %X on cooling. The 1st cooling DSC trace should be the same (after normalizing for the weight of the sample) for all the treated samples since the melt had 300,000 times the value of the reptation time to erase any thermo-mechanical influence on the structure. The large scatter of values for %X (from 23% to 47%) does not make sense unless one accepts the idea that a melt can assume many states, which can be out of equilibrium or in equilibrium. The use of such a concept as the longest relaxation time to characterize the thermodynamic state of a melt seems to be at best incomplete (see Note 1).
Now, if a melt can assume many non-equilibrium states it can only be due to the population of b-grains since the crystals have all melted. We should be able to observe the evolution of such non equilibrium states by following the heat capacity, Cp(T), which can be extracted from the normalized heat flow in DSC after calibration (baseline+sapphire). If the melt of a polymer can be made truly unstable, its heat capacity should be different from the thermodynamic value at that temperature, describing its equilibrium state, and it should evolve in time towards this equilibrium state. The theoretical Cp(T) below and above Tg of polymers is well known, calculated from IR measurements and from other considerations, and their value is tabulated in many reference handbooks. Hence, it is easy to determine the “Excess Cp” at each temperature by subtracting the Cp theoretical value from the Cp(T) of the data. If the Excess Cp is positive there is an excess of free volume, i.e. a lack of b-grains with respect to its equilibrium value, and when it is negative we have an excess of b-grains. Graph 2 below shows the excess Cp(T) for the 1st and the 2nd heating curve for a sample which has been Rheo-Fluidified and then annealed in a vacuum oven at T=145oC.
Graph 2
The 1st heating trace (pink) shows an excess of b-grains even persisting in the melt region (beyond the melting peak for T>255) whereas the heat capacity for the 2nd heating trace (yellow) returns to its equilibrium value (excess=0).
In conclusion, it’s fair to say that until one can reconcile the discrepancy shown in the accounting of crystallinity in DSC (Graph 1) and the variation of Cp(T) with thermo-mechanical history, one cannot affirm to understand the thermodynamics and the kinetics of crystallization in polymers.
Wanna learn more?
[i] “Physics of Polymer Interactions: a New Approach. Applications to Rheology and Processing”, Book HANSER (to be published 2018). https://sites.google.com/a/eknetcampus.com/books-and-essays/
