Thermal Analysis of Polymeric Materials

The second method makes use of data on mass loss, collected in a series of different constant-heating-rate experiments as outlined in Fig. 4.198 for polystyrene. Isoconversion occurs at different temperatures for different heating rates. As one reaches the point of isoconversion, the integrated, mass-dependent functions f(p) must be identical. Thus, one has again achieved information about k(T) at different temperatures. Naturally this analysis is only valid if the kinetics has not changed over the range of temperature and conversion.

The Arrhenius equation of Appendix 7 (Fig. A.7.2) can be inserted in the general

The constant heating rate q is dT/dt and can be used to change time into temperature and obtain:

Integrating to constant conversion p gives, for the left-hand side, a constant value, while the right-hand side is somewhat more difficult. Either, one carries out numerical integrations, or it may be sufficient to notice that the logarithm of the integral portion in the center can be approximated as a linear function of 1/T so that one arrives at the final equation:

log q2 – log q1 –0.4567(Ea/R)[(1/T2) – (1/T1)]

The prediction of the lifetime of polystyrene in Figs. 4.198 seems to give similar activation energies for both, the parameter-jump method and the isoconversion method. It is of interest to judge how precise this lifetime prediction is. The error introduced by two standard deviations, to reach a 95% confidence level, gives for the steady-state parameter-jump method an estimate ranging from 2.7×108 to 1.3×109 years at 300 K, an error of about 80%. One can see from this estimate that relatively small errors in activation energy can produce large errors in the lifetime.

Polystyrene is, however, a relatively ideal case. In Fig. 4.199, data on the isoconversion of decomposition of a segmented polyurethane are reproduced. Only the first 10% of decomposition in vacuum give parallel lines with activation energies between 145 and 170 kJ mol 1. Perhaps it may still be possible to describe this early decomposition as a single process. At higher conversion, the kinetics is much more complex and no interpretation is possible with thermogravimetry alone.

Summary of Chapter 4. At the end of this discussion of thermal analysis tools it may be worthwhile to attempt a brief summary. The basic theory of thermal analysis is well represented by macroscopic equilibrium and nonequilibrium thermodynamics, and the connection to the microscopic description is given by statistical thermodynamics and kinetics. All of these theories are highly developed, but they have not been applied to their fullest in the description of materials. The reason for this failure to

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Fig. 4.199

develop thermal analysis into a well-rounded field of scientific enquiry is not necessarily a lack of precision in measurement, but rather the fact that these theories were developed before many of the present-day measuring tools were available. Traditional courses in physical chemistry and engineering often do not contain the material needed to understand the application of thermal analysis, but do treat topics that unnecessarily complicate the subject. In turn, many thermal analyses are carried out at less than optimum precision, since the operators are often unaware of the added information that could be obtained and used for interpretation. On the other hand, as shown in the discussion of thermogravimetry, data can also easily be over-interpreted because of a lack of detailed understanding of the microscopic processes. Finally, it should be remarked that the field of thermal analysis goes far beyond the basic techniques described here and is constantly growing. Almost any measurement which can be done at different temperatures can be expanded into thermal analysis, and any series of thermal analysis techniques can be combined for valuable multiple-parameter measurements. With the understanding of the basic techniques presented here, one should be able to expand one’s knowledge to the practically unlimited opportunities of thermal analysis.

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References

General References

Sect. 4.1. Much information on temperature can be found in the proceedings of Symposia on Temperature, sponsored by the American Institute of Physics, the Instrument Society of America, and the National Institute for Standards and Technology. Also, see the treatise: by various eds Temperature: Its Measurement and Control in Science and Industry. Vol 1, 1944; Vol 2, 1955; Vol 3, 1962; Reinhold, New York. Vol 4, 1972 73, Inst Soc America, Pittsburgh; Vol 5, 1982; Vol 6, 1992, Am Inst Physics, New York.

General references on temperature measurement are: Schooley JF (1987) Thermometry. CRC Press, Boca Raton, FL; Quinn TJ (1983) Temperature. Academic Press, New York; Sturtevant J M (1971) Temperature Measurement. In Weissberger A, Rossiter BW, Physical Methods of Chemistry. Wiley-Interscience, New York, Vol I, Part V, Chap I.

For a general introduction into metrology see, for example: Klein AH (1974) The World of Measurement. Simon and Schuster, New York.

Thomson GW, Douslin DR (1971) Determination of Pressure and Volume. In Weissberger A, Rossiter BW, Physical Methods of Chemistry. Wiley–Interscience, New York, Vol I, Part V, Chap II.

A Series of biannual reviews in the journal Anal Chem covers all thermal analysis: Wendlandt WW (1982) 54: 97R; (1984) 56: 250R. (1986) 58: 1R; Dollimore D, 60: 274R; (1990) 62: 44R; (1992) 64: 147R; (1994) 66: 17R; (1996) 68: 63R; Dollimore D, Lerdkanckanaporn S, (1998) 70: 27R; Dollimore D, Phang P (2000) 72: 27R; Vyazovkin S (2002) 74: 2749; (2004) 76: 3299.

Sect. 4.2. Basic textbooks for this section are: Wunderlich B (1990) Thermal Analysis. Academic Press, Boston; Hemminger W Höhne G (1984) Calorimetry. Verlag Chemie, Weinheim, Germany.

Some classical texts on calorimetry are: White WP (1928) The Modern Calorimeter. Chem Catalog Co, New York. Swietoslawski W (1946) Microcalorimetry. Reinhold Publ, New York. Sturtevant JM (1971) Calorimetry. In: Weissberger A Rossiter BW, eds, Techniques of Chemistry Vol I, Part V. Wiley-Interscience, New York.

Tables of calorimetric data are found in: a. Landolt–Boernstein (1956–71) Zahlenwerte und Funktionen. Springer, Berlin, 6th ed, Vol II, Parts 1 5; continued as Hellwege KH, ed. New Series Group IV: Macroscopic and Technical Properties of Matter. b. Am Pet Inst Research Proj 44 Selected Values of Properties of Hydrocarbon and Related Compounds. Thermodynamics Res Center, Texas A&M University, College Station, TX. c. Touloukian YS, Ho CY, eds (1970–1979) Thermophysical Properties of Matter, The TPRC Data Series. Vols 4 6, Specific Heat. IFI/Plenum, New York, NY. d. NBS Technical Notes 270–3 to 270–7 (1968–1973) Selected Values of Chemical Thermodynamic Properties. Institute for Basic Standards, National Bureau of Standards, Washington, DC. e. Karapet'yants MKh, Karapet'yants ML (1970) Thermodynamic Constants of Inorganic and Organic Compounds. Humphrey Sci Publ, Ann Arbor. Translated by Schmork J. f. Lide DR (annual editions) CRC Handbook of Chemistry and Physics. CRC Press, Boca Raton, FL.

Sect. 4.3. The main reference for DSC of Materials is the two-volumes compendium: Turi E (1997) Thermal Characterization of Polymeric Materials. Academic Press, San Diego.

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Books documenting the history and principles of DTA and DSC are: Smothers WJ, Chiang Y (1966) Handbook of Differential Thermal Analysis. Chem Publ, New York; Mackenzie RC (1970,1972) Differential Thermal Analysis, Vols 1 and 2. Interscience Publ, New York; Wunderlich B (1971) Differential Thermal Analysis. In Weissberger A, Rossiter BW, Physical Methods of Chemistry, Vol 1, Part V, Chap 8. Wiley, New York; Wendlandt WW (1986) Thermal Analysis, 3rd edition. Wiley-Interscience, New York.

The earliest DTA-related research papers are: LeChatelier H (1887) Z Phys Chem 1: 396; Roberts-Austen WC (1899) Metallographist 2: 186; Kurnakov NS (1904) Z anorg Chem 42: 184; Saladin E (1904) Iron and Steel Metallurgy Metallography 7: 237; see also: LeChatelier H (1904) Rev Met 1: 134.

A number of Reference DTA and TGA curves can be found in: Liptay G (1971–76) Atlas of Thermoanalytical Curves, Vols 1 5. Heyden, London; Sadtler (1965) DTA Reference Thermograms, Vols 1 7. Sadtler, Philadelphia, PA.

Newer books on DSC are: Wunderlich B (1990) Thermal Analysis. Academic Press, Boston; Höhne G, Hemminger W, Flammersheim HJ (2003) Differential Scanning Calorimetry, 2nd edn. Springer, Berlin; Brown ME, Gallagher PK, Cheng SZD, eds (1998, 2002) Handbook of Thermal Analysis and Calorimetry. Vol 1, Principles and Practice, Vol 3, Applications to Polymers and Plastics. Elsevier Science, Amsterdam.

For information about the identification of the glass transition see: Seyler RJ, ed (1994) Assignment of the Glass Transition. Am Soc Testing and Materials, Philadelphia; the analysis of polymer melting is discussed in Wunderlich B (1980) Macromolecular Physics, Vol 3, Crystal Melting. Academic Press, New York.

Sect. 4.4. A general review of TMDSC is found in: Reading M, ed (2004/5) Basic Theory and Practice for Modulated Temperature Differential Scanning Calorimetry. Kluwer, Dordrecht. The Proceedings of the last five Lähnwitz Seminars in 1996, 1998, 2000, 2002, and 2004, focusing on TMDSC, discussed authoritatively: (1997) Temperature-modulated Calorimetry. Thermochim Acta Vol 304/305; (1999) Investigation of Phase Transitions by Temperaturemodulated Calorimetry. Ibid Vol 330; (2001) Frequency and Time Dependent Heat Capacity. Ibid Vol 377; (2003) Thermodynamics and Calorimetry of Small Systems. Ibid Vol 403; (2005) Thermodynamics and Calorimetry of Thin Films. Ibid, to be published. For early publications see: Reading M, Hahn BK, Crowe BS (1993) Method and Apparatus for Modulated Differential Analysis. US Patent 5,224,775, July 6. Reading M, Elliot D, Hill VL (1993) J Thermal Anal 40: 949; Wunderlich B, Jin Y, Boller A (1994) Thermochim Acta 238: 277; Boller A, Jin Y, Wunderlich B (1994) J Thermal Anal 42: 307.

Sect. 4.5. Wendland WW (1986) Thermal Analysis, 3rd ed. Wiley, New York; Gallagher PG (1997) Chap 1 in Turi E, ed. Thermal Characterization of Polymeric Materials. Academic Press, San Diego; Barth HG, Mays JW (1991) Modern Methods of Polymer Characterization. Wiley-Interscience, New York.

For the measurement of length, pressure, and volume, see: Thomson GW, Douslin DR, (1971) Determination of Pressure and Volume. In Weissberger A, Rossiter BW, eds. Physical Methods of Chemistry, Vol 1, Part V. Wiley–Interscience, New York. See also Riga AT, Neag CM, eds (1991) Materials Characterization by Thermomechanical Analysis. ASTM STP 1136, American Society for Testing and Materials, Philadelphia, 1991.

For further study of DMA see, for example: Ferry JD (1980) Viscoelastic Properties in Polymers, 3rd edition. J. Wiley, New York; Ward IM (1983) Mechanical Properties of Solid Polymers, 2nd edn. Wiley, New York; Meier DJ (1978) Molecular Basis of Transitions and Relaxations. Gordon and Breach, New York; McCrum NG, Read BE, Williams G (1967) Anelastic and Dielectric Effects in Polymeric Solids. Wiley, New York; Aklonis JJ, MacKnight WJ (1967) Introduction to Polymer Viscoelasticity. Wiley, New York; Matsuoka S (1992) Relaxation Phenomena in Polymers. Hanser Publ, Munich.

For dielectric measurements, see for example: Karasz FE, ed (1972) Dielectric Properties of Polymers. Plenum Press, New York; Blythe AR (1979) Electrical Properties of Polymers.

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Cambridge University Press, Cambridge. A related technique, that of thermally stimulated current analysis is described by Ibar JP (1993) Fundamentals of Thermal Stimulated Current and Relaxation Map Analysis. SLP Press, New Canaan.

Sect. 4.6. Descriptions of thermogravimetry are given by: Gallagher PG (1997) Chap 1 in Turi E, ed. Thermal Characterization of Polymeric Materials. Academic Press, San Diego; Wunderlich B (1990) Thermal Analysis. Academic Press, Boston; Duval C (1963) Inorganic Thermogravimetric Analysis, 2nd ed. Elsevier, Amsterdam. (For decomposition of polymers see Refs to Sect 3.4).

Early thermogravimetry: Agricola G (1556) De re metallica. Translated from the 1556 edition by Hoover HC, Hoover LH (1912) Dover Publ, New York; Hannay JB (1877) J Chem Soc 32: 381; Ramsey W (1877) ibid 395; Honda K (1915) Sci Rep, Tohoku Univ 4: 97.

For tables of the integrated functions and general discussions of calculations of TGA, see Doyle, CD (1966) Quantitative Calculations in Thermogravimetric Analysis. In Slade PR, Jr, Jenkins LT, eds. Techniques and Methods of Polymer Evaluation. Marcel Dekker, New York, pp 113–216.

Typical TGA and DTA curves are collected by: Liptay L (1971–76) Atlas of Thermoanalytical Curves, Vols 1 5. Heyden, London.

For lifetime determination, the jump method is described by: Flynn JH, Dickens B (1976) Thermochim Acta 15: 1–16. The isoconversion method, by: Ozawa T (1965) Bull Chem Soc, Japan 38: 1881–1886.

Specific References

1.The Proceedings of the International Conferences on Thermal Analysis (and Calorimetry) can be found in book form under the title: Thermal Analysis. Various publ and edts, 1965, 1969, 1972, 1975, 1977, 1980, 1982. More recent proceedings are published in: Thermochim Acta (1985) 92/93; (1988) 133/135; J Thermal Anal Cal (1993) 40, (1997) 49, (2001) 64.

2.Proceedings of the annual NATAS Conferences, changing eds, for example: . Kociba KJ, Kociba BJ, eds (2002) Proc. 30th NATAS Conf in Pittsburgh, PA, Sept 23–25, vol 30. Since 2003 the proceedings are issued on CD.

3.The ITS 90 was initiated on Jan 1, 1990 and is described by Preston–Thomas H, Quinn TJ (1992) The International Temperature Scale of 1990: Parts I and II. In: Murray TP, Shepard RL, eds (1992) Temperature: Its Measurement and Control in Science and Industry, Vol 6, Part 1. Am Inst Physics, New York pp 63–74. See also Preston–Thomas H (1990) Metrologia 27: 3. See there also for the conversion of the IPTS 68 and earlier scales to the ITS 90.

4.Lavoisier AL (1789) translated by Kerr R (1790) Elements of Chemistry. Part III, Chap III. Edinburgh. Frequently reprinted, for example, printed as a facsimile: (1965) Dover Publications, New York.

5.For the original paper on the ice calorimeter, see: Bunsen R (1870) Ann Phys 141: 1.

6.Updyke J, Gay C, Schmidt HH (1966) Improved Precision Ice Calorimeter. Rev Sci Instr, 37: 1010–1013.

7.Southard JC (1941) A Modified Calorimeter for High Temperatures. The Heat Content of Silica, Wollastonite and Thorium Dioxide above 25°. J Am Chem Soc 63: 3142–3146.

8.Sunner S, Manson M (1979) Experimental Chemical Thermodynamics, Vol 1, Combustion Calorimetry. Pergamon, Oxford, 1979.

9.Nernst W (1911) Der Energieinhalt fester Stoffe. Ann Phys 36: 395–439; see also Lindemann FA, Koref F, Nernst W, (1910) Untersuchungen an specifischen Wärmen bei tiefen Temperaturen. I and II. Sitzber kgl preuss Akad Wiss 12(13): 247–292.

10.Tasumi M, Matsuo T, Suga H, Seki S (1975) Adiabatic Calorimeter for High-resolution Heat Capacity Measurements in the Temperature Range from 12 to 300 K. Bull Chem Soc, Japan 48: 3060–3066.

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11.Oetting FL, West ED (1982) An Adiabatic Calorimeter for the Range 300 to 700 K. J Chem Thermodynamics 14: 107–114.

12.Chang, SS (1976) A Self-balancing Nanovolt Potentiometer System for Thermometry and Calorimetry. J Res Natl Bur Stand 80A: 669–675.

13.Gmelin E, Rödhammer P (1981) Automatic Low Temperature Calorimetry for the Range 0.3 320 K. J Phys E, Instrument 14: 223–238.

14.Tian A (1933) Researches on Calorimetry. Generalization of the Method of Electrical Compensation. Microcalorimetry. J Chim Phys 30: 665–708; and Calvet E (1948) Compensated Differential Microcalorimeter. Compt rend 226:1702–1704.

15.Palermo E, Chiu J (1976) Critical Review of Methods for the Determination of Purity by Differential Scanning Calorimetry. Thermochim Acta 14: 1–12.

16.Moros SA, Stewart D (1976) Automated and Computerized System for Purity Determination by Differential Scanning Calorimetry. Thermochim Acta 14: 13–24.

17.Sarge SM, Bauerecker S, Cammenga HK (1988) Calorimetric Determination of Purity by Simulation of DSC Curves. Thermochim Acta 129: 309–324.

18.Plato C, Glasgow AR, Jr (1969) Differential Scanning Calorimetry as a General Method for Determining the Purity and Heat of Fusion of High-purity Organic Chemicals. Application to 95 compounds. Anal Chem 41: 330–336 (1969).

19.Wunderlich B, Jin Y (1993) Thermal Properties of the Four Allotropes of Carbon. Thermochim Acta 226: 169–176.

20.Jin Y, Wunderlich B (1991) The Heat Capacity of n-Paraffins and Polyethylene. J Phys Chem 95: 9000–9007.

21.Watson ES, O’Neill MJ, Justin J, Brenner N (1964) Differential Scanning Calorimeter for Quantitative Differential Thermal Analysis. Anal Chem 36: 1233–1238.

22.Gill PS, Sauerbrunn SR Reading M (1993) Modulated Differential Scanning Calorimetry. J Thermal Anal 40: 931–939.

23.Wunderlich B (1987) Development Towards a Single-Run DSC for Heat Capacity Measurement. J Thermal Anal 32: 1949–1955.

24.Jin Y, Wunderlich B (1993) Single-run Heat Capacity Measurement by DSC: Principle, Experimental and Data Analysis. Thermochim Acta 226: 155–161.

25.Jin Y, Wunderlich B (1990,1992) Single Run Heat Capacity Measurements. J Thermal Anal 36: 765–789; II. Experiments at Subambient Temperatures. Ibid 36: 1519–1543; III. Data Analysis. Ibid 38: 2257–2272.

26.Höhne G, Hemminger W, Flammersheim, HJ (2003) Differential Scanning Calorimetry, 2nd edn, Sect 5.4. Springer, Berlin

27.Lau SF, Suzuki H, Wunderlich B (1984) The Thermodynamic Properties of Polytetrafluoroethylene. J Polymer Sci, Polymer Phys Ed 22: 379–405.

28.Mathot VBF, Pijpers MFJ (1989) Heat Capacity, Enthalpy, and Crystallinity of Polymers from DSC Measurements and Determination of the DSC Peak Baseline. Thermochim Acta 151: 241–259.

29.Wunderlich B, Androsch R, Pyda M, Kwon YK (2000) Heat Capacities by Multifrequency Saw-tooth Modulation. Thermochim Acta 348: 181–190.

30.Moon I, Androsch R, Wunderlich B (2000) A Calibration of the Various Heatconduction Paths for a Heat-flux-type Temperature-modulated DSC. Thermochim Acta 357/358: 285–291.

31.Androsch R, Moon I, Kreitmeier K, Wunderlich B (2000) Determination of Heat Capacity with a Sawtooth-type, Power-compensated Temperature-modulated DSC. Thermochim Acta 357/358: 267–278.

32.Androsch R, Wunderlich B (1999) Temperature-modulated DSC Using Higher Harmonics of the Fourier Transform. Thermochim Acta 333: 27–32.

33.Pak J, Wunderlich B (2001) Heat Capacity by Sawtooth-modulated, Standard Heat-flux Differential Scanning Calorimeter with Close Control of the Heater Temperature. Thermochim Acta 367/368: 229–238.

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34.Kwon YK, Androsch R, Pyda M, Wunderlich B (2001) Multi-frequency Sawtooth Modulation of a Power-compensation Differential Scanning Calorimeter. Thermochim. Acta 367/368: 203–215.

35.Pyda M, Kwon YK, Wunderlich B (2001) Heat Capacity Measurement by Sawtooth Modulated Standard Heat-flux Differential Scanning Calorimeter with Sample-temperature Control. Thermochim Acta 367/368: 217–227.

36.Wunderlich B (1997) Modeling the Heat Flow and Heat Capacity of Modulated Differential Scanning Calorimetry. J Thermal Anal 48: 207–224.

37.Merzlyakov M, Wurm A, Zorzut M, Schick C (1999) Frequency and Temperature Amplitude Dependence of Complex Heat Capacity in the Melting Region of Polymers, J Macromolecular Sci, Phys 38: 1045–1054.

38.Toda A, Tomita C, Hikosaka M (1998) Temperature Modulated DSC of Irreversible Melting of Nylon 6 Crystals. J Thermal Analysis 54: 623–635.

39.Wunderlich B (2003) Reversible Crystallization and the Rigid Amorphous Phase in Semicrystalline Macromolecules. Progr Polym Sci 28: 383–450.

40.Wunderlich,, B, Bodily, DM, Kaplan MH (1964) Theory and Measurement of the Glasstransformation Interval of Polystyrene. J Appl Phys 35: 95–102.

41.For a series of publications on the glass transitions of polystyrene and poly(ethylene terephthalate) see: Modulated Differential Scanning Calorimetry in the Glass Transition Region, written by: Thomas LC, Boller A, Kreitmeier S, Okazaki I, Wunderlich B (1997) J Thermal Analysis 49: 57–70; Thermochim Acta 291: 85–94; (1996) J Polymer Sci, Part B: Polymer Phys 34: 2941–2952; J Thermal Analysis 47: 1013–1026; Thermochim Acta 284: 1–19.

42.van Mele B, Rahier H, van Assche G, Swier S (2004) The Application of Modulated Temperature Differential Scanning Calorimetry for the Characterization of Curing Systems. In Reading M, ed, Basic Theory and Practice for Modulated Temperature Differential Scanning Calorimetry. Kluwer, Dordrecht, The Netherlands, pp 72–152.

43.Schmieder K, Wolf K (1952) The Temperature and Frequency Dependence of the Mechanical Properties of Some High Polymers. Kolloid Z 127: 65–78.

44.Wurm A, Merzlyakov M, Schick C (2000) Reversible Melting During Crystallization of Polymers Studied by Temperature Modulated Techniques (TMDSC, TMDMA). J Thermal Anal Calorimetry 60: 807–820; see also: (1998) Reversible Melting Probed by Temperature Modulated Dynamic Mechanical and Calorimetric Measurements. J Colloid Polymer Sci 276: 289–296.

45.Schmieder K, Wolf K (1953) Mechanical Relaxation Phenomena in High Polymers. Kolloid Z 134: 149–189.

46.Duval C (1951) Continuous Weighing in Analytical Chemistry. Anal Chem 23: 1271–1286.

47.Details on the TGA of Figs. 4.177–180 are described in: Wiedemann HG (1964) Thermogravimetric Investigations. VI. Universal Device for Gravimetric Determinations under Variable Conditions. Chemie Ing Tech 36: 1105–1114.

48.Zitomer F (1968) Thermogravimetric Mass Spectrometric Analysis. Anal Chem 40: 1091–1095.

49.Paulik F, Paulik J, Erdey L (1958) The “Derivatograph.” I. An Automatic Recording Apparatus for Simultaneously Conducting Differential Thermal Analysis, Thermogravimetry, and Derivative Thermogravimetry. Z anal Chem 160: 241–252. For standardization, quasi-isothermal and isobaric analyses and some example research with the Derivatograph see also: (1966) Anal Chim Acta 34: 419–426; Paulik F, Paulik J (1973) J Thermal Anal 5: 253–270; (1975) 8: 557–576.

50.Sørensen OT, Rouquerol J, eds (2003) Sample-controlled Thermal Analysis (SCTA): Orign, Goals, Multiple Forms, Applications, and Future. Kluwer, Amsterdam.

CHAPTER 5

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Structure and Properties of Materials

In this Chap. 5 of the book on Thermal Analysis of Materials, the link between microscopic and macroscopic descriptions of crystals is given in Sects. 5.1 3. This is followed by a thermodynamic analysis of melting of crystals and isotropization of mesophases in terms of entropy and enthalpy in Sects. 5.4 and 5.5. The final section deals with the properties of liquids and glasses (Sect. 5.6).

5.1 Crystal Structure

5.1.1 Introduction

Crystals have always fascinated man. To this day crystals are not only treasured because of their beauty [1], but there remain some of the ancient beliefs of possible magic. Huygens’ drawing of a CaCO3 crystal in Fig. 2.101 was first prepared to explain the occurrence of the birefringence more than 100 years before Dalton’s proof of the existence of atoms (see Sect. 1.1.1). Even earlier suggestions exist of links between the external regularity of crystals, depicted in Fig. 3.77, and the atomic structure. Kepler analyzed in 1611 the hexagonal structure of snow flakes, as shown in Fig. 5.1 in terms of regularly packed balls, and Hooke (1665) used bullets to understand the shapes of diamonds. The correct NaCl structure could be derived 100 years ago purely from packing consideration of spheres of different radius and charge, as reproduced in Fig. 5.2. All leads to the conclusion that motifs must repeat themselves regularly in space. Today we know about the details of the structure of the motifs and their arrangement in the crystals through X-ray diffraction, as can be seen in Fig. 5.81, below, and electron microscopy, as illustrated in Fig. 2.102.

To characterize a crystal, one must describe the motif and find the repetition scheme that generates the crystal. There are an unlimited number of different motifs, i.e., represented by atoms, ions, parts of molecules, molecules, or even groups of many molecules, but for the resulting crystals, there are only a limited number of repetition schemes. They are represented by the 230 space groups. The motifs and repetitionschemes in crystals and helices are discussed in Sects. 5.1.1–6.

The combination of motifs and repetition schemes is the second topic of this section. The space group of the lattice is the scheme that acts on motifs placed into the unit cell and generates the crystal structure. The space group is determined by X- ray diffraction or, less frequently, by electron or neutron diffraction. Typically one to 18 repeating units of a macromolecule exist within one unit cell, but helices with long translational repeats may place many more repeating units into one unit cell. For example, poly(m-methylstyrene)s with a 2*40/11 helix form a four-chain unit cell with

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Fig. 5.1

Fig. 5.2

160 repeating units, compared to the similar poly(o-methylstyrene) of Fig. 5.29, below, which has the similar 2*4/1 helix and 16 repeating units. Central for the understanding of a crystal structure is the packing of its motifs.

The closest packing without distorting motifs will usually yield the most stable crystals with lowest free enthalpy, as discussed in Sects. 5.1.7 and 5.1.8. The packing