Study of Polymer Glasses by Modulated Differential Scanning

In the Laboratory

Study of Polymer Glasses by Modulated Differential Scanning Calorimetry in the Undergraduate Physical Chemistry Laboratory

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J. C. W. Folmer and Stefan Franzen* Department of Chemistry, North Carolina State University, Raleigh, NC 27695-8204; *[email protected]

Thermodynamics and calorimetry have long been an important part of the undergraduate physical chemistry curriculum. Differential scanning calorimetry (DSC) is a workhorse technique in research as well as in industry, particularly in the polymer field (1). Its educational value has also long been recognized (2–7). The advent of the modulated form of the technique, MDSC (8, 9), further strengthens the position of thermal analysis. It also presents educational opportunities and challenges. For example, a major advantage of the technique is that it can determine which part of the heat flow responds to the imposed modulation reversibly and which part does not. This division strongly benefits the study of polymers because it helps to unravel the often complicated phase transitions in these materials. MDSC also offers an excellent educational approach to the concept of reversibility. Unfortunately, many students only acquire a superficial understanding of this key concept of classical thermodynamics in the current curriculum. In our opinion this is largely due to the fact that the concept is taught on a purely abstract level. The great thermodynamicists of the 19th century developed the concept of reversibility by considering a hypothetical process that moves through a succession of states-of-rest. Although rigorous in its definition, this abstract path represents a difficult educational approach for many students. A practical application of the concept, for example as an experiment in a physical chemistry laboratory course, could do much to reinforce and deepen the students’ understanding. To this end we propose an MDSC experiment involving a polymer glass in comparison to a semicrystalline polymer. It facilitates the introduction of the distinction between first- and second-order phase transitions and explores the border between thermodynamics and kinetics by considering the importance of timescales. The experiment complements and augments a standard experiment, the adiabatic compression of a gas, by comparison. It also exposes the students to a technique they may well encounter later and thus underlines the relevance of thermal analysis as a widespread application of modern thermodynamics. Modulated DSC In classical DSC experiments the temperature is scanned; that is, it is raised at a fixed rate dT兾dt = β (T denotes temperature, t represents time). A combination of computer control and proper design ensures a linear ramp to a good approximation

T (t ) = T (0 ) + β t

(1)

The temperature can typically range from ᎑100 ⬚C to 300 ⬚C. Under these conditions heat will flow into both the

sample pan and an empty reference pan predominantly by conduction. Ignoring thermal resistivity, the heat flow dq兾dt into an object is related to its temperature change dT兾dt by its heat capacity dq dT = Cp dt dt

(2)

As Cppan + Cpsample > Cppan the heat flow into the empty reference pan is generally smaller. The difference in heat flow dq兾dt induces a small temperature difference ∆T between sample and reference across a material with known (calibrated) heat resistance K. The temperature difference ∆T is measured. We can relate this signal to the heat flow using Newton’s law of heat flow. There is a close analogy of this law to Ohm’s law of electrical flow ∆T = K

dq ; dt

(Newton)

∆V = R

dQ = RI dt

(Ohm)

(3)

It is useful to note the analogy of heat flow to charge flow, that is the electrical current I = dQ/dt. Newton’s law is the thermal equivalent of Ohm’s law of electrical flow. Although the former is older, the latter is more familiar to the students. The ∆T signal can be related to the heat capacity Cp of the sample by ∆T = K

dq dT = K Cp = KCp β dt dt

(4)

Equation 4 implies that larger signals can be achieved for the same Cp (i.e., the sensitivity S increased) by increasing the heating rate β. In principle, the heat capacity of the sample should be easy to measure. Unfortunately eq 4 is an incomplete description of the DSC process. In general, there is an additional so-called kinetic term f (t,T ) that gives rise to instrumental broadening of the measured signal ∆T = KCp

dT + f (t ,T T) dt

(5)

Once the heating rate has been constant for a sufficiently long time, the kinetic term reduces to a constant or weakly linear term. Under the conditions of stationary heat flow a moreor-less flat baseline is obtained and the following equation results

∆Tstationary ≈ KCp β + a (+bT + ...)

JChemEd.chem.wisc.edu • Vol. 80 No. 7 July 2003 • Journal of Chemical Education

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In the Laboratory

The heat capacity Cp and the constant term a can be obtained by varying β, but this requires multiple scans. A sudden change in Cp, however, should be visible as a sudden change in the baseline. In practice, a sigmoidal shape is typically observed as a result of the broadening by the f (t,T ) term. The broadening is more severe for higher heating rates β. This reduces resolution because two subsequent events will tend to broaden into one. Thus sensitivity can only be bought at the expense of resolution, as both depend on β. In the modulated version of the technique (MDSC), computer control over the temperature is used to superpose a relatively fast temperature oscillation on the linear ramp. By specifying values for β, α, and ω the instrument is programmed to impose the following temperature function

T (t ) = T (0) + βt + α sin (ωt )

(7)

To a good approximation, the resulting heating rate will oscillate as dT = β + αω cos (ω t ) dt

(8)

The heating rate dT兾dt fluctuates between β − αω and β + αω. This oscillation has three advantages. Firstly, resolution and sensitivity are uncoupled because the sensitivity depends on dT兾dt, which no longer depends on the scan rate β alone, but also on αω. Low values of β (even zero) can be used to obtain data with high temperature resolution. Secondly, the heat capacity can be determined in a single MDSC scan because the experiment is essentially done over a range of heating rates at once. The focus of our work is the third advantage. The heat flow dq兾dt into the sample has two components: a gradual trend and an oscillating part (see Figures 5 and 6). The trend provides the total heat flow (as in DSC), but the amplitude of the oscillation presents the advantage of a second signal. It contains additional information, closely linked to the timescale of the oscillation. Therefore MDSC provides an opportunity to elucidate the importance of timescales. Reversing versus Nonreversing Heat Flow Equations 1–7 all involve time. Therefore timescales are inherent to scanning calorimetry, but they are not part of the classical thermodynamics taught in undergraduate classes. For example, reversibility is either defined by demanding that a process can be reversed by an arbitrarily small change in the conditions (T, P, etc.) or by saying that the process progresses along a series of equilibrium states. Neither definition involves timescales: it is simply assumed that there is plenty of time to achieve such a process. In practice this is often not so, but students are seldom taught how to deal with this explicitly. A possible remedy is to review the concept of reversibility from a perspective of timescales in a simple thought experiment. Consider two beakers containing a liquid connected with a siphon. If the siphon’s diameter is large and the viscosity of the liquid low, one of the beakers can be moved up and down rather quickly and the fluid levels will readjust to their joint equilibrium level on a timescale that is fast compared to that of the imposed changes. The system is quick to respond in a reversible way. The transport process 814

can indeed be reversed by an infinitesimal change in height δh. If however the siphon is very narrow (or the viscosity large), much longer timescales are needed to achieve reversibility. This simple example clarifies the need to couple the concepts of timescales and reversibility by defining the terms reversing and nonreversing as follows. A process is •

reversing if the system responds in a reversible way on the timescale of the experiment (or faster), or

•

nonreversing if the system is either too slow to respond reversibly on the timescale of the experiment or if it is irreversible altogether (on any timescale).

Adopting these terms, which are in regular use in current MDSC applications, the siphoning process is said to be driven into a nonreversing limit when done too fast. The classical concept of reversibility can be represented as a limit for time going to infinity in the thought experiment. Of course, it can be argued that we have tacitly crossed the border from equilibrium into nonequilibrium thermodynamics or even into kinetics with the above argument. In our opinion, deliberately investigating this border has considerable educational merit. After all, timescales are an issue in many other fields, NMR to name but one. Without a proper discussion of the issue one could even question the legitimacy of DSC in our curriculum. Sometimes the timescale for which a reversible process is driven into nonreversing behavior cannot be attained in an experiment. For example, the adiabatic compression of a gas can only be driven into nonreversing behavior if the compression is faster than the speed of sound. If done by hand (a standard experiment done in many labs), this is not achievable as gas molecules respond very rapidly. However, other processes such as the vitrification of a polymer are much slower. We propose to study such a process by MDSC, because both reversing and nonreversing effects can be demonstrated in the same experiment. The extra information obtained from the oscillation amplitude of the ∆T signal yields the reversing part of the heat flow. By subtraction from the total heat flow the nonreversing part can be obtained. A comparison to the adiabatic compression of gas experiment strengthens the educational value of the latter. At the same time a bridge is created to kinetics experiments, typically also part of the undergraduate lab curriculum. Vitrification versus Crystallization Students are usually familiar with the process of solidification into a crystalline material, for example water into ice, but are less familiar with the process of vitrification, for example of a polymeric liquid into a glass. This lack of familiarity is unfortunate. Glasses play an increasing role in daily life. They are encountered not only as the traditional windows or beakers but also in many polymers and amorphous semiconductor applications. Even in biopolymers, for example in protein folding trajectories, there is growing recognition of glasslike behavior (10–13). There are also educational reasons to consider glass transitions in greater depth in the curriculum. There is a widespread misconception that glasses are simply ‘undercooled liquids’ that needs to be addressed. In reality both vitrification and crystallization are a form of solidification in which

Journal of Chemical Education • Vol. 80 No. 7 July 2003 • JChemEd.chem.wisc.edu

In the Laboratory

Figure 1. A schematic diagram for the enthalpy and heat capacity as a function of temperature for a glass transition. (A) The slope of the enthalpy vs temperature plot changes at the glass transition temperature. This results in a sudden change in the temperature-derivative of enthalpy, i.e., the heat capacity. (B) The heat capacity is shown for an instantaneous change (solid line) and with inclusion of a kinetic term as indicated in eq 3 (dashed line) as observed in DSC.

the molecules of the liquid lose most of their freedom of translational and rotational movement. These movements are still possible in a liquid, although they are strongly coupled as opposed to independent as in an ideal gas. In either type of solidification eventually only vibrational modes remain. The atoms can only move around their equilibrium positions. This change represents a large loss in accessible degrees of freedom and thus corresponds to a change in heat capacity ∆Cp (Figure 1). Because Cp = (∂H兾∂T)P, the slope of the enthalpy H(T ) should change at the transition temperature (Figure 1A). For a glass transition a rapid change in Cp(T ) (Figure 1B) at the glass transition temperature Tg is therefore all we would expect. However, in the event of crystallization there is an additional effect: the molecules become ordered. This represents a sudden loss of entropy ∆S. As the process comes to equilibrium at the melting temperature Tm, this translates into a latent heat of crystallization (or fusion) ∆Hfus = Tm∆S, on top of the change in heat capacity, as shown in Figure 2. Vitrification (Figure 1) does not have this complication, which actually makes it a simpler process to explain. As is explained below, simple diagrams such as Figures 1 and 2 have considerable educational value because they facilitate pointing out that enthalpy and entropy are both firstorder derivatives of the Gibbs free energy function G and that the heat capacity is a second-order derivative of G: G T H = − 1 ∂ T ∂

S = −

Cp = T

Figure 2. A schematic diagram for the enthalpy and heat capacity as a function of temperature during a crystalline melting event. (A) The latent heat of the phase transition is manifest as a sudden change in the enthalpy, ∆Htransition. The heat capacity also changes upon phase transition. (B) The heat capacity change upon phase transition results in a change in Cp. The heat capacity is infinite at the phase transition since the enthalpy function is discontinuous there (solid line). In DSC the inclusion of a kinetic term results in a shift and smoothing of the observed heat capacity change upon transition (dashed line).

∂S ∂T

∂G ∂T

P

(9) P

= −T P

∂ 2G ∂T 2

P

A glass transition is typically observed as a shift in the baseline in DSC (ref 1; see dashed line in Figure 1B). It reflects the change in the heat capacity ∆Cp—a second-order derivative. Crystalline fusion at the melting point Tm is more complicated because there is an additional jump in enthalpy and entropy—first-order derivatives. Consequently, the second-order derivative has a singularity at Tm. The term ƒ(t,T ) broadens the sharp singularity in Cp into a measurable DSC peak (see dashed line in Figure 2). From the surface area of the peak ∆Hfus can be obtained by numerical integration. Ehrenfest introduced a distinction between first- and second-order phase transitions. His classification is based on identifying the lowest-order derivative of G that displays a discontinuity at the transition temperature. It is mentioned in many textbooks (e.g., Atkins in ref 14 ). The classification assumes a discrete transition temperature and does not discuss timescales. Within this scheme, crystallization is a firstorder phase transition. Vitrification can be compared to a second-order phase transition, but only if we postulate that the ∆Cp change is sudden and that the base line shift is smoothed into a sigmoidal shape by the kinetic term ƒ(t,T ) in eq 5. However, this postulate does not necessarily hold


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Figure 3. A schematic diagram of the influence of the timescale on a glass transition. (A) Rapid cooling can lead to formation of glass that is not in its lowest energy state. Fast cooling is represented as the dashed enthalpy line, whereas slow cooling is represented by the solid line. (B) The observed phase transition will be shifted owing to rapid cooling as shown by the first derivative of the enthalpy (i.e., the heat capacity plots for fast cooling [dashed line] and slow cooling [solid line], respectively).

true. Vitrification can be so gradual that its change ∆Cp does not represent a true discontinuity. The formation of a glass is truly a time-dependent phenomenon. A proper description would involve the time-temperature superposition principle (15). Unfortunately, this leads beyond the simplicity of the Ehrenfest classification, as well as the scope of undergraduate education. In polymer science it is customary to approach this educational problem by first postulating that there is indeed a discrete ∆Cp discontinuity (16) at a discrete temperature Tg and treating time-temperature superposition as a later refinement. We think this approach is justified provided the groundwork for the later refinement is laid by invoking a proper discussion of timescales. Role of Timescales in Glass Transitions If a glass system is heated and cooled at the same rate, the same sigmoidal signal is observed upon heating or cooling; that is, the system shows a reversing response. However, if the system is heated and cooled more slowly, the system goes into a different state, as slower cooling allows a more thorough loss of degrees of freedom. The glass transition point Tg is said to shift to lower temperatures upon slower cooling as shown in Figure 3. Figure 3 shows a connection between time and transition temperature that provides one experimental approach to the time-temperature superposition of polymer theory. If the heating and cooling rates are not the same, the system is driven into a nonreversing limit. A good example of such behavior is seen when the system is allowed to age, that is allowed to equilibrate at a temperature not too far un816

Figure 4. A schematic diagram of the influence of aging on a glass transition. (A) A rapidly cooled glass can be aged as shown by the arrow. (B) The appearance of the phase transition in this case can be similar to a first-order phase transition due to crystallization.

der Tg (Figure 4A). The system relaxes to a state of lower enthalpy as it would have done upon slower cooling. Subsequent fast heating will cause the system to overheat up to the ‘fast’ Tg, where it will quickly ‘recover’ the missing enthalpy ∆Hrecovery. In conventional DSC this is seen as an endothermic ‘recovery’ peak superposed on the sigmoidal baseline shift that symbolizes ∆Cp. As Figure 4B shows, the result may look deceptively like a crystalline melt curve. In MDSC the two effects can be separated, because recovery is a nonreversing effect, whereas the underlying sigmoid is a reversing one. The reason is that the system is studied at two timescales at once. The underlying ramp rate b produces a trend line with the total heat flow shown in Figure 5. As in classical DSC, it reflects the system’s total response at the timescale of the overall experiment, including its thermal history. The amplitude of the heat flow oscillations, however, reflects the reversing heat flow at a timescale of 1yw. This reversing signal is typically sigmoidal and allows us to observe Tg at this particular timescale. Its characteristic shape tells us that we are indeed dealing with a glass transition. By subtraction of the reversing signal from the total trend we obtain the nonreversing component from which we can measure ∆Hrecovery by integration. This value depends on the thermal history of the sample. Safety Considerations The experiments described require a TA Instruments Q100 modulating DSC system with propane, ethane, and propylene driven refrigeration cooling system, allowing scans from ᎑100 °C to 400 °C to investigate polyethylene glycol (PEG) and polypropylene glycol (PPG) or other similar models. Neither the equipment nor the materials used pose any


In the Laboratory

Figure 5. Representation of modulated heat flow and its decomposition. The bottom solid line represents the trend line, which is the average of the raw data as a function of temperature (or time). After subtraction of the trend line, the amplitude of the oscillation can be determined (see top solid line). The raw data and analysis shown here were obtained on polypropylene glycol sample (PPG) discussed in the laboratory example and shown in Figure 6.

Figure 6. (A) The total heat capacity for a PPG glass transition. (B) The resolution of the total heat capacity into a sigmoidal reversing part (bold curve) and a nonreversing part (dashed curve). The latter shows the peak due to the enthalpy of recovery. The in part negative values result from the fact that instrumental factors may enter into the nonreversing signal.

particular safety problems. The materials are in widespread use, for example in cosmetic products and as inert components in drug tablets. Lab Experiment Two related polymers, PEG and PPG, are investigated. Both polymers are readily available (SIGMA). Although their chemical structures only differ by the additional methyl group in the propylene glycol monomer, the physical properties of the two polymers are very different. Atactic polypropylene glycol PPG is a viscous liquid that undergoes a glass transition ca. ᎑70 °C. PEG is a semicrystalline material that melts at ca. 60 °C. In the case of PPG, a Hamilton syringe is used to fill a pan with approximately 10 µL of material and the student is asked to do a run from T= ᎑90 ⬚C to ᎑45 ⬚C with a heating rate of β = 5 K兾min, a modulation of amplitude α = 0.5 K, and a frequency of ω = (2π兾60) s᎑1. The sample is allowed to age at T = ᎑90 ⬚C for 5 minutes. This whole procedure takes about 45 minutes. Typical results are shown in Figure 6. The reversing part of the heat flow shows that telltale sigmoidal shape of a glass transition, from which the students are asked to determine the glass transition temperature and the size of ∆Cp. From the nonreversing part they are asked to determine the ∆H of recovery. In the case of PEG, approximately 3–5 mg of powder are heated from 25 ⬚C to 85⬚ using the same values of β, α, and ω. This also takes about 45 minutes total. Typical results are shown in Figure 7. It is quite clear that the reversing part is not a sigmoid, confirming that this is not a glass transition. The students are asked to determine the ∆H of fusion by peak integration and calculate ∆S of fusion from ∆H and Tm.

Figure 7. (A) The (apparent) total heat capacity of PEG during the melting endotherm. The integrated value over the peak is 207.8 J/g, the onset temperature is 52.04 °C, and the peak temperature is 58.24 °C. (B) The reversing and nonreversing heat capacity for the same thermogram as shown in panel A. Note the nonsigmoidal shape of the reversing signal.

The students are asked to draw schematic H(T) versus T diagrams (Figures 1 and 2) inserting the values for Tm, Tg, ∆Cp, ∆H, and so forth, that they have determined. In a typical afternoon lab it is feasible to ask the students to do a few repeat runs, for example three PPG runs with different modulation frequencies. For further details refer to the Supplementary Material.W


817

In the Laboratory

Conclusion A MDSC experiment on vitrification offers interesting educational opportunities for the undergraduate physical chemistry laboratory because it highlights the concept of reversibility in relation to the timescale of an experiment. The relationship between kinetics and thermodynamics can be introduced at an appropriate level for undergraduate students. The MDSC experiment on a glass material is best contrasted to a material involving a crystalline melting event. The comparison of crystallization and vitrification facilitates a better understanding of phase transitions. The students gain practical experience with an important technique and are introduced to concepts that complement and reinforce what they learn in a lecture course. Supplemental Material Student instructions in a slide-presentation format are available in this issue of JCE Online. W

Literature Cited 1. Hoehne, G.; Hemminger, W.; Flammersheim, H. J. Differential Scanning Calorimetry. An Introduction for Practitioners; Springer: Berlin, 1996.

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2. Chowdhry, B.; Leharne, S. J. Chem. Educ. 1997, 74, 236. 3. Temme, S. M. J. Chem. Educ. 1995, 72, 916. 4. Vebrel, J. l.; Grohens, Y.; Kadmiri, A.; Growling, E. W. J. Chem. Educ. 1993, 70, 501. 5. Choi, S.; Larrabee, J. A. J. Chem. Educ. 1989, 77, 864. 6. Crandall, E. W.; Pennington, M. J. Chem. Educ. 1980, 57, 824. 7. Brown, M. E. J. Chem. Educ. 1979, 56, 310. 8. Gill, P. S.; Sauerbrunn, S. R.; Reading, M. J. Therm. Anal. 1993, 40, 931–939. 9. Simon, S. L. Thermochimica Acta 2001, 374, 55–71. 10. Angell, C. A. Science 1995, 267, 1924–1935. 11. Sokolov, A. P. H.; Grimm, H.; Kahn, R. J. Chem. Phy. 1999, 110, 7053–7057. 12. Onuchic, J. N.; Wolynes, P. G. J. Chem. Phy. 1993, 98, 2218– 2224. 13. Frauenfelder, H.; Wolynes, P. G. Science 1985, 229, 339–346. 14. Atkins, P. Physical Chemistry, 6th ed.; W.H. Freeman & Co.: New York, 1997. 15. Ferry, J. D. Viscoelastic Properties of Polymers; John Wiley & Sons: New York, 1980. 16. Tool, A. Q. J. Am. Cer. Soc. 1946, 29, 240–252.


Study of Polymer Glasses by Modulated Differential Scanning

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