1. Fundamentals of Polymer Science and technology
University of Shanghai for Science and Technology
Hua Zou

2. Table of Contents Topic Classes Introduction to Polymer Science 3
Polymer Synthesis 18
Solid-State Properties 9
Viscoelasticity
Polymer Degradation and Environment
Polymer Solutions
Miscellaneous 6
In Total 45

3. Lecture 9 Thermal Transition and Properties
9.1 Fundamental Thermodynamic Relationships
9.2 Measurement Techniques
9.3 Structure-Property Relationships
9.4 Effect of Molecular Weight, Composition, and Pressure on Tg

4. Lecture 9.1 Fundamental Thermodynamic Relationships

5. 9.1 Fundamental Thermodynamic Relationships
Many of the commonly used techniques to determine Tg and Tm can be understood on the basis of the thermodynamic definition of a phase transition originally proposed by Paul Ehrenfest in 1933.
First-Order Transition
is defined as one for which a discontinuity occurs in the first derivative of the Gibbs free energy (G).
According to the first law of thermodynamics for a reversible, closed system, the Gibbs free energy can be expressed in differential form as a function of temperature and pressure, G(T, p), as
S is entropy and V is the system volume.

6. 9.1 Fundamental Thermodynamic Relationships
The Gibbs free energy
The free energy may be differentiated with respect to pressure (at constant temperature) as
In terms of describing transitions in polymer systems, this indicates that a first-order transition should occur as a discontinuity in volume.
Volume is easily measured as a function of temperature by a technique called dilatometry. The dependence of volume on temperature in the region about the crystalline-melting temperature approximates such a transition.
Figure 4-9 Thermodynamic first-order transition in volume at constant pressure.

7. 9.1 Fundamental Thermodynamic Relationships
Second-Order Transitions
The glass transition approximates an Ehrenfest second-order transition.
This means that a discontinuity should be observed in the second derivatives of the Gibbs free energy.
Three possible second derivatives can be used to provide a basis for the experimental measurement of Tg.

8. 9.1 Fundamental Thermodynamic Relationships
Second-Order Transitions
Since entropy is not an experimentally measurable quantity, eq. (4.13) may be recast into a more useful form in terms of the specific heat at constant pressure, Cp, which is defined as
From the first law of thermodynamics
Figure 4-10 Thermodynamic second-order transition in specific heat at constant pressure.
A second-order transition should occur as a discontinuity in specific heat. Specific heat is easily measured by calorimetric techniques such as DSC.

9. 9.1 Fundamental Thermodynamic Relationships
Second-Order Transitions
The glass transition is not a true thermodynamic transition but, rather, it is considered to be a pseudo-second-order transition that is influenced by the kinetics of glass formation (i.e., the rate of heating or cooling).
Both volume and specific-heat data for polymers closely approximate second-order transition behavior, but the discontinuities or changes in slope are more gradual and are affected by the heating rate.

10. Lecture 9.2 Measurement Techniques

11. 9.2 Measurement Techniques
A wide variety of experimental methods can be used to determine Tg and Tm in polymers
refractive index,
NMR line width, and
birefringence
dilatometry
differential scanning calorimetry (DSC）
dynamic-mechanical analysis
dielectric spectroscopy
modulus in tensile, stress relaxation, and other mechanical tests
able to detect low-temperature
secondary relaxations

12. 9.2 Measurement Techniques: Dilatometry
Dilatometry: one of the earliest methods
A small sample of polymer is sealed in a glass bulb to which a precision-bored, calibrated glass capillary is attached.
Mercury, whose coefficient of thermal expansion is accurately known, is used to fill the bulb and part of the capillary.
The dilatometer is then immersed in a controlled-temperature bath and the height of the mercury in the capillary is recorded at different temperatures.
Heating rate is normally kept very small to assure thermal equilibrium, especially near Tg.
The specific volume of the polymer sample can be obtained as a function of temperature.

13. 9.2 Measurement Techniques: Dilatometry
Figure Dilatometric data of specific volume of a semicrystalline polymer, poly(N,N’-sebacoyl piperazine), plotted against temperature.

14. 9.2 Measurement Techniques: Dilatometry
Table 4-6. Dilatometric Data for Some Representative Polymers
The change in thermal-expansion coefficient going from the liquid (i.e., T > Tg) to the glassy state

15. 9.2 Measurement Techniques: DSC
DSC, is a thermal analysis technique that looks at how a material’s heat capacity (Cp) is changed by temperature. A sample of known mass is heated or cooled and the changes in its heat capacity are tracked as changes in the heat flow.
This allows the detection of transitions such as melts, glass transitions, phase changes, and curing.
keep the temperature of the two equal

16. 9.2 Measurement Techniques: DSC
When the sample undergoes a physical transformation such as phase transitions, more or less heat will need to flow to it than the reference to maintain both at the same temperature.
Whether less or more heat must flow to the sample depends on whether the process is exothermic(放热的) or endothermic(吸热的).

17. 9.2 Measurement Techniques: DSC
A typical DSC plot of a polymer
As a solid sample melts to a liquid it will require more heat flowing to the sample to increase its temperature at the same rate as the reference. This is due to the absorption of heat by the sample as it undergoes the endothermic phase transition from solid to liquid.
As the sample undergoes exothermic processes (such as crystallization) less heat is required to raise the sample temperature.

18. 9.2 Measurement Techniques: Heat-Distortion Temperature
The heat distortion temperature is determined by the following test procedure outlined in ASTM D648.
The test specimen is loaded in three-point bending in the edgewise direction. The outer fiber stress used for testing is either 0.455/1.82 MPa, and the temperature is increased at 2 °C/min until the specimen deflects mm.
Limitations: the sample is not thermally isotropic and, thick samples will contain a temperature gradient.
Amorphous polymer: HDT is slightly lower than the Tg.Semicrystalline polymer: more close to Tm

19. 9.2 Measurement Techniques: Heat-Distortion Temperature
Table 4-7. Thermal-Transition Temperatures

20. Lecture 9.3 Structure-Property Relationships

21. Lecture 9.3 Structure-Property Relationships
Table 4-2. Glass-Transition Temperatures of Some Amorphous Polymers

22. Lecture 9.3 Structure-Property Relationships
Table 4-3. Thermal Transitions of Some Semicrystalline Polymers

23. Lecture 9.3 Structure-Property Relationships
Both Tg and Tm are strongly influenced by the chemical structure of the repeating unit.
In general, both Tg and Tm increase with decreasing flexibility of the polymer chain.
Flexibility decreases with increasing aromatic composition of the main chain or by incorporation of bulky substituent groups or nonrotational (e.g., unsaturated) groups in the main chain.
Table 4-8. Effect of Backbone Structure of the Crystalline-Melting Temperature
of Polyesters Derived from Ethylene Glycol (HOCH2CH2OH)

24. Lecture 9.3 Structure-Property Relationships
Chain flexibility is particularly important in determining Tg.
Flexible chains, as may be obtained by incorporating an oxygen atom into the main chain (e.g., polydimethylsiloxane), are capable of large-scale molecular motions at very low temperatures and, therefore, have low Tg.
Bulky substituent groups hinder chain rotation and therefore raise Tg as shown by structure–Tg comparisons for several vinyl polymers.

25. Lecture 9.3 Structure-Property Relationships
For comparably sized substituent groups, increasing polarity, which may enhance intermolecular interactions, can elevate Tg.
This is illustrated by Tg data for the vinyl polymers.
Table 4-9. Glass-Transition Temperatures of Selected Vinyl Polymers

26. Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg

27. Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Molecular-Weight Dependence
Composition Dependence
Pressure Dependence
Effect of Heating Rate
Effect of Crosslinks

28. Molecular-Weight Dependence
Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Molecular-Weight Dependence
The glass-transition temperature increases with molecular weight at low molecular weight but reaches a point at moderate molecular weight where further increase in molecular weight has very little effect on Tg.
The crystalline-melting temperature, Tm, follows a similar dependence on molecular weight. The particular molecular-weight average most relevant to Tg is the number-average.
This dependence can be rationalized on the basis of the free-volume theory of the glass transition.
Larger free volume is associated with the ends of long polymer chains than with other chain segments, free volume increases with an increasing number of chain ends (i.e., decreasing molecular weight).

29. Molecular-Weight Dependence
Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Molecular-Weight Dependence
The form of dependence of Tg on molecular weight is approximated by the Fox–Flory equation
Tg∞: limiting value of Tg at high molecular weight (obtained from the intercept of a plot of Tg versus reciprocal number-average molecular weight)
K: a constant for a given polymer.
Equation has been found to give a good fit of experimental data for many polymers; however, there is evidence that K may not be constant for molecular weights below about 10,000.

30. Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Molecular-Weight Dependence
Table Fox–Flory Parameters

31. Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Composition Dependence.
When a second component, either a low-molecular-weight additive or a second polymer, is blended to form a homogeneous mixture, the Tg of the mixture will depend upon the amount of each component and upon the Tg of the second component.
The form of the Tg–composition dependence may be approximated by several theoretical or semiempirical models.
An approximate relationship between the Tg of a miscible mixture and composition is given by the simple rule of mixtures, which for a binary mixture is given as
W1 is the weight fraction and Tg,i (in Kelvins) is the glass-transition temperature of the ith component (i.e., component 1 or 2 in a binary mixture).

32. Composition Dependence.
Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Composition Dependence.
For a multicomponent mixture, we can write
The simple rule of mixtures is a good approximation for blends of two or more polymers but overestimates the Tg of polymers plasticized with a low-molecular-weight organic compound such as an ester or phthalate.

33. Composition Dependence.
Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Composition Dependence.
Kelley-Bueche equation, derived from the isofree volume model.
f fractional free volume fg
αf thermal expansion coefficient
ϕ volume fraction
when T= Tg, f = fg, αf,1(T-Tg,1)ϕ1 + αf,2(T-Tg,2)ϕ2 = 0

34. Composition Dependence.
Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Composition Dependence.
Fox equation: an empirical equation based on thermodynamic theory
Another commonly used empirical relation that can be derived from eq. (4.32) is the logarithmic rule of mixtures given as

35. Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Pressure Dependence
Compared to effects of molecular weight and plasticization, Tg is relatively insensitive to pressure.

36. Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Effect of Heating Rate
The Tg has a small dependence on the heating or cooling rates in DSC and other methods of thermal characterization.
Samples that are slowly heated through the glass transition exhibit a lower Tg than those that are rapidly heated due to the non-equilibrium state of the glass.
The relationship between Tg (K) and the heating rate, q (K min-1), is given in the form where a and b are polymer-specific parameters (e.g., a = K and b = 4.02 K for PS; a = 383 K and b = 4.23 K for PMMA).

37. Lecture 9.4 Effect of Molecular Weight, Composition, and Pressure on Tg
Effect of Crosslinks
As a result of the restriction of long-range segmental motion, crosslinking elevates Tg.
The form of the relationship between Tg and cross-link density is often given by the Fox–Loshaek equation
is the glass-transition temperature of the crosslinked polymer,
is the glass-transition temperature of the uncrosslinked polymer,
kc is a polymer-specific constant,
ρc represents the number of crosslinks per gram.