Fundamentals of Thermal Analysis and Differential Scanning Calorimetry

1. FUNDAMENTALS OF THERMAL ANALYSIS AND DIFFERENTIAL SCANNING CalorimetryApplication in Materials Science Investigations Analiza cieplna i kalorymetria różnicowa w badaniach materiałów Tomasz Czeppe Lecture 2 (3). Thermal properties of materials, temperature scalesand thermometry

2. Introduction All old and modern methods of thermal analysis require precise and repeatable temperature measurements and for this thermometers had to be invented. Before short discussion concerning thermometry two basic properties related to precision of the temperature measurements, thermal expansion and thermal conductivity are discussed. Thermal conductivity is the basic event for all thermal analysis methods and phase transitions, thermal expansion is also the subject of dilatometry and thermo-mechanical analysis TMA.

3. 2.1. Thermal expansion The source for the thermal expansion is an asymmetry in the atoms thermal oscillations that is anharmonicpart in the oscillations. As we must remember,on the base of quantum physics the energy of atomicthermal oscillations in the solid state are quantified revealing both character of the elastic waves and particles named phonons. To understand the nature of the thermal expansion in the solid state, at first it is enough to analyse dependence of the potential Ep and kinetic Ek components of the total energy Et from the distance L in case of the simple two atomic, linear molecule (Fig. 2.1).

4. 2.1. Thermal expansion As was schematically marked at large distances L of the atoms the interaction is very week so the force F is 0 and the Ep very small and constant. And some smaller distance the interaction starts to be effective generating the attracting force F. If the distance L further decreases the repulsive forces are increasing what generates the equilibrium distance L0, at which F=0 and the potential energy Ep has a minimum. Further decreasing the distance L cases strong increase in the repulsion forces. As is clear there is strong asymmetry in attracting and repulsive interaction between the atoms in a molecule. Fig. 2.1. Schematic presentation of the dependence of Ep and F from the atoms distance L in the binary molecule.

5. 2.1. Thermal expansion Quantitatively, the dependence of Ep = f(L) may be presented in a simple way as: 2.1.1 Where: B1, B2- constants > 0, and n1 < m2 . This type of expression has name Lenard-Jones potential. In general there is no physical meaning of such form except for metals,in which the first, attractive part for n1=1 may be interpreted as Coulomb potential between positive ions and collective electrons, while the second part - the repulsive energy caused by the collective electron movements. As long the vibrations are small the repulsive and attracting force is symmetrical around the equilibrium point (harmonic vibrations: F~L), but while the temperature and energy increases the amplitude increases and asymmetry of the both type of force manifests in changing the position of the equilibrium point L0as is shown in Fig. 2.2.

6. 2.1. Thermal expansion The basic equation for the linear coefficient of thermal expansion awas introduced by Fermi and Frenkel: 2.1.2. Where: k - Boltzmann constant, l- anharmonic coefficient, t- quasi-elastic coefficient. There is many other formulas. One of them connects a with dissipation energy Edis and n1 and m2 parameters of Lennard-Jones potential for the linear molecule but is good also for interpretation of experimental results for many materials (inspite of simplicity): Fig. 2.2. Schematic presentation of the deviation of L0 from the original position due to large amplitude of vibrations in an an-harmonic case in the binary molecule. Blue line concerns Ep in the case of harmonic vibrations.

7. 2.1. Thermal expansion • This conclusion are in a good agreement with the following formula for the thermal coefficient of the volume expansion b (Grüneisen relation): • 2.1.4. • Where g is constant and B - volume elastic moduli (B-1 is compressability), Cv-heat capacity with constant volume. • Both V and B are only weakly temperature dependent so in a result: • b ~ Cv that is atlow temperatures b ~ T3 and at high temperatures b ~ constant.

8. 2.1. Thermal expansion Another important relation between volume expansion and free energy F: 2.1.5 Supplies very important information, as F is an additive function also b must be additive and may be calculated by summarization of different components. So, we may divide b in parts related to the crystalline lattice (phonons), electron gas and magnetic particles (magnons): b = bf + be + bm Similarly to the heat capacity, at low temperatures (T < 0,1.qD) be becomes equally important as bfand increases going to 0 temperature, at higher temperatures T > 0,1.qD the bf predominates. The magnetic part bmfor ferromagnetic phases may be in the same range as bf and be.

9. 2.1. Thermal expansion • Anisotropy of the volume expansion • In case of amorphous and polycrystalline materials the isotropy of thermal expansion may be assumed. In the case of monocrystals, in the case of crystal lattice not belonging to the cubic symmetry, the volume expansion is not isotropic. In this case heating of the crystal results in anisotropic dimensional increase which must be represented by the tensor of deformation Eij. For the temperature increase by DT the tensor components proportionally change in the chosen direction : • 2.1.6. eij= aijDT • Both eij and aijare symmetrical tensors of the second order. If we transform the tensor to the main axis of the crystal lattice, only diagonal components are different from 0.

10. 2.1. Thermal expansion Fig. 2.3. Anisotropic change of the crystal lattice dimensions by the tensor of thermal expansion in main axis system in 2D. Anisotropic change of the crystal lattice dimensions in the 2D main axis system is presented in Fig. XX. For the 3D system the volume coefficient of thermal expansion will have the form: 2.1.7.b = a11 + a22 + a33

11. 6n a2 a1 a1 2.1. Thermal expansion • Summarising: • Linear coefficient of thermal expansion a is reversibly proportional to dissipation energy; • Volume coefficient of thermal expansion b is proportional to Cv and may be summarized from the components: lattice, electron and other; • In case ofmonocrystals: • for the cubic crystals : a11 = a22 = a33 = a and b = 3a isotropic expansion; • for the hexagonal and trigonal crystals: a11 = a22 = a1,a33 = a2 and b = 2a1+ a2 • anisotropic expansion (contraction) in one direction; • for rhombohedral crystals: b = a11 + a22 + a33anisotropic expansion in 3D • In all this system the ellipsoid axis are parallel to the crystallographic axis, but for monoclinic and triclinic crystals things are more complicated.

12. 2.2. Thermal conductivity • Determination of the thermal conductivity requires investigation of the space and time dependence of temperature T = f(x, y, z, t), that is thermalfield. While the temperature reveals time dependence, dT/dt=0, we have stationary thermal field, opposite, while dT/dt=f(t) the field is nonstationary. • Interconnection of all the points revealing the same temperature T in (x, y, z) space defines an isothermal surface. • Increase of the temperature in the direction perpendicular to the isothermal surface changescharacter from the scalar quantity of T to the vector quantity of the stress of the thermal fieldW: • 2.1.8W = grad T= (dT/dx + dT/dy + dT/dz) • The amount of heat passing in a unit of time t through the unit area of the isothermal surface S in the direction x: • 2.1.9 q= (dT/dt)S-1.x • is named density of heat flux(vector character) (Fig.2.4)

13. 2.2. Thermal conductivity Fig. 2.4. Basic definitions concerning temperature stress field and heat flux.

14. 2.2. Thermal conductivity Fourier law: Fourier suggested a proportionality between density of heat fluxq and the thermal fieldW: 2.1.10.q = lW = -l grad T where l names thermal conductivity coefficient. In not-stationary case q depends also on time, and the following equation for thermal conductivity equation for the thermal body applies: Where a - coefficient of the temperature conductivity, r- density.

15. 2.2. Thermal conductivity • The next case takes into account also the sources of heat operating in the thermal body: • Where means the heat source power, d, c are constants. • To know non-stationary thermal field in the body the above differential equations must be sold. This requires also the knowledge of such named “boundary conditions” at the starting point of time of the process: T (x, y, z, 0)=const. • As the boundary conditions define also physical situations which may happened in thermal analysis, they are summarized below: • temperature distribution is determined at the surface: Tsurf.(t) = f(t), in the stationary case Tsurf.(t)=const.; • heat flux qis determined at the surface: qsurf.(t)= f(t), in the stationary caseqsurf.(t) = const.; • a stationary case of the heat exchange by convection: qsurf.= nT(Tsurf. -Tspace), where Tspace means temperature of the surrounding space and nT – heat exchange coefficient.

16. 2.2. Thermal conductivity Heat conductivity in dielectrics: In dielectrics heat conductivity is determined by the quantified elastic vibrations named phonons. In the case of hard dielectrics thermal conductivity coefficient ldepends on heat capacity of the gas of phonons: Where cph is average heat capacity of the gas of phonons and means average free path of the phonons, dependent on the diffraction on the phonons and defects of the structure, means an average speed of phonons. The average free path of the phonons depends on the relation of the sample temperature T to the Debay temperature TD : in case T >>TDCph is not temperature dependent,; and depends on the phonon=phonon interaction, in case T <<TD and and strongly increases but is limited by the size of the sample.

17. 2.2. Thermal conductivity Heat conductivity in metals: In metals thermal conductivity coefficient l has two components, from phonons lph, and from gas of electrons le. While le>>lph: Where e- electron charge, electron conductivity. As results: l/reT= const. remains constant what is known as Wiedemann-Franz-Lorentz law. The ferromagnetic transition causes a jump in thermal conductivity. In many cases lin metals is proportional to the external pressure P, increasing with the increase of the external pressure.

18. 2.3. Temperature scales The basic scale of temperature accepted in science is kelvin scale. It represents the absolute temperature scale basing on gas thermodynamics lows with two main points: absolute 0 and the triple point of waterarbitrary placed at 273.16 K. This range defines also the measure of 1 K. This is such named primary temperature scale, however in the experimental practice the highest precision in temperature determination cannot be achieved. So, there is need for the secondary temperature scales. This scale is named International Temperature Scale 1990 (ITS 1990). It bases on the 14 temperature points for thermometers scaling. The points are triple, melting or freezing points of gasses, water and metallic elements at constant pressure P=101,325 Pa at standard atmosphere.

19. 2.3. Temperature scales • The ITS 1990 scale starts from the triple point of H2at equilibrium (13,8033 K - 259,3467oC), • the low temperature range limit is at the triple point of water at 273,16K - 0,02oC, and this gives constant difference between the C and K scales, • high temperature range starts from Ga melting point at 302.9146 K - 29.7646oC and finishes at the freezing point of Cu (1357.77 K - 1084.62oC). • Why the freezing, that is crystallization temperatures were chosen for the thermometers calibration will be clear when we see thermal curves including melting and crystallization of pure metals. • To use temperature scale the thermometer had to be invented. • The temperature ranges covered by the different (basic)thermometers are as follows: • 0.65 – 5 K vapor-pressure relationships for 3He and 4He; • 3.0 – 24.5561 K helium gas thermometer; • 13.8033 – 1234.93 K platinum resistance thermometer; • Above 1234.93 monochromatic radiation pyrometer.

20. 2.4. Thermometers and thermometry • Frequently used types of thermometers are : • Liquid in glass thermometers • Resistance thermometers and thermocouples • 2.4.1. Liquid in glass thermometers • Till the development of electronic resistance thermometers and semiconductor thermometers a Hg post in thin glass or quartz capillary was the most useful and widely applied thermometer. However, its development, fundamental for the development of sciences, was not easy, due to the lack of the repeatability and precision of measurements. (The same concerned also barometers and similar measuring apparats.) This was caused by the lack of the technology of the glass capillary production. • The temperature range of application is 240-800 K. The common precision is ±0.001 K, but may be higher.

21. 2.4. Thermometers and thermometry • Frequently used types of thermometers are : • Liquid in glass thermometers • Resistance thermometers and thermocouples • 2.4.1. Liquid in glass thermometers • Till the development of electronic resistance thermometers and semiconductor thermometers a Hg post in thin glass or quartz capillary was the most useful and widely applied thermometer. However, its development, fundamental for the development of sciences, was not easy, due to the lack of the repeatability and precision of measurements. (The same concerned also barometers and similar measuring apparats.) This was caused by the lack of the technology of the glass capillary production. • The temperature range of application is 240-800 K. The common precision is ±0.001 K, but may be higher.

22. Err = kn(Tt – T0) n- Hg column above the bath (e.g-84 K) Ts - bath temperature Tt – temperature read from the therm. (e.g -366.0 K) T0 – thermometer scale starting temperature (eg.273) k- differential of glass-Hg expansivity k ~ 1.6.10-4 K-1 Ts = 366 +1.6.10-4.84(366-273) = 366.0 + 1.25=367.25 Tt n 2.4. Thermometers and thermometry Ts • The proper temperature determination needs: • Direct contact with measured object • Time enough for thermal equilibrium • Corrections for heat flow • The source of errors could be: • Lackof the stable volume of the bulb. • This may result from the thermal history of the thermometer • or from the hysteresis of the volume, • Thermometer lag, • Pressure effect, • Immersion effect (Fig. 2.5)

23. 2.4. Thermometers and thermometry To have high precision temperature measurements the liquid in glass thermometers must be calibrated for one temperature from time to time. The hysteresis results from the difference in time required for the achieving the final volume of the bulb in heating (short) and in cooling. The last one may be very long. Thermometer lag results from the thermal conductivity of the thermometer materials, that is propertime is required for heat to enter and equilibrated the internal part of thermometer. Pressure differences in the bulb results in differences in reading temperature. They may result from the different position of thermometer. The most important errors results from the immersion problems. Some thermometers have marks informing how deep must be immersion in the liquid.

24. 2.4. Thermometers and thermometry • Other type of thermometers • Quartz thermometer – bases on the measurements of frequency of an oscillator controlled by the quartz crystal, temperature dependent. Temperature range 200-500 K, resolution 10-4 K; • Pyrometer – measures light intensity of a given, narrow frequency range; • Bimetallic thermometer – not very precise; • Vapour pressure thermometer, measures pressure under a liquid being in contact with the investigated system. Especially useful at low temperatures; • Gas thermometer- uses Boil-Marriott gas low PV=nRT; • Noise thermometer- measures random noise of electrons in conductors;