4.3 Thermal Transitions and Properties

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4.3.1 Fundamental Thermodynamic Relationships

4.3 Thermal Transitions and Properties

• first order transition:

- defined as one for which a discontinuity occurs in the first derivatives of G - the Gibbs free energy (G) is a function of T and p,

- from eq.(4.12), a first order transition should occur as a discontinuity in V (Figure 4-9)

- dilatometry: measure volume change as a function of T

( 4 . 1 0 ) -

- - V d p S d T

d G   

(4.12) -

- -

(4.11) -

- -

p V G T S G

T p

 



 









 



 







First Order Transitions

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• second order transition:

- defined as one for which a discontinuity occurs in the second derivatives of G - three possible derivatives useful to determine T_g,

- eq.(4.13), entropy is not measurable quantity, so use specific heat instead,

- combining eq.(4.17) into eq.(4.13); a second order transition should occur as (4.15)

- - - (4.14) -

- -

(4.13) -

- -

2 2

p p T

T T

p p

T V p

G T

p V p

G

T S T

G



 







 











 

 









 







 



 









 







 



 







 

Second Order Transitions

(4.17) -

- -

(4.16) -

- -

p p

T T S

C

T C H



 







 



 







 

discontinuity in the slope of V as a function of p

discontinuity in the slope of V as a function of T

(4.18) -

- 1 -

p T

V V 

 







 



 





 

(4.19) -

- 1 -

T p

V

V 



 







 



 



 



compressibility

thermal expansion coefficient

Figure 4-11A Figure 4-11B

V d p T d S

d H  

p

p T

T S T

H 



 







 



 







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- the discontinuities in C_p, , and  occur at the second order transition,

- glass transition: pseudo-second order transition

dependent on the kinetics (heating or cooling rate) discontinuities or changes in slope are gradual

(4.22) -

- -

(4.21) -

- -

(4.20) -

- -

1 2

1 , 2

,





















 C

_p

C

_p

C

_p

Figure 4-11A Figure 4-11B Figure 4-10

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 transition: state change caused by temperature (or pressure) change

 thermodynamic 1^st order transition:

- 1^st derivatives of Gibbs free energy are discontinuous ex) entropy(S), volume(V)

- phase change during transition

ex) crystalline melting (fusion), crystallization, boiling

 thermodynamic 2^nd order transition:

- 1^st derivatives of Gibbs free energy are continuous, but 2^nd derivatives of Gibbs free energy are discontinuous ex) heat capacity (C_p), thermal expansion coefficient (α) - single phase transition (no phase change during transition)

ex) glass transition

Thermodynamic Transitions

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- glass transition (T

_g

): beginning of chain segmental motion occur in the amorphous region during heating

glassy state rubbery state

- thermodynamic second

order transition (kinetic process) - rate dependent process

- crystalline melting (T

_m

): fusion of crystalline during heating

crystalline melt

- thermodynamic first

order transition (equilibrium process) - rate independent process

T>T

_g

T>T

_m

Thermal Transitions of Polymers

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T

_g

T

_m

equilibrium non-equilibrium

specific volume

(cc/g)

relax. time > exp. time relax. time ≤ exp. time

G_u = H_u – TS_u = 0 at T = T_m(= T_m^o) - for a semicrystalline polymer,

temperature (K)

equilibrium

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4.3.2 Measurement Techniques

• thermal transitions:

- can be detected by measuring refractive index change with T NMR line (peak) width

birefringence - most common techniques: dilatometry

differential scanning calorimetry (DSC) dynamic mechanical analysis (DMA) dielectric analysis (DEA)

- DMA & DEA can detect secondary relaxations that occur below T_g

- observed in modulus vs. T curves obtained from mechanical tests such as tensile and stress relaxation tests

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Dilatometry

• composed of a glass capillary connected with a bulb filled with mercury

• heating rate (1~2 ^oC/min) should be low to assume thermal equilibrium

• measures specific volume of sample vs. T, Figure 4-12

• T_g – the T at which V-T curve changes its slope (discontinuity in )

• T_m – discontinuity or step change in V

• T_g and (thermal expansion coefficient) values of some polymers, Table 4-6

• the change in  at T_g from liquid (rubbery) (T>T_g) to glassy state (T<T_g),

•  increases with decreasing T_g of polymers as shown in Table 4-6

• according to Simha and Boyer,

(4 .2 3 ) -

-

g

-

l



  



(4.24) -

- 113 - .

0 T

g



 

Fig. 10-9 Fig. 10-10

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ref) 김성철 외, 고분자공학I, 희중당, 1994

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Differential Scanning Calorimetry

• one of the most widely used methods, Figure 4-13 - polymer sample weight = 10 ~ 30 mg

- sample and reference pans are heated individually to maintain both at the same T - measures differential power for both pans as a function of T

- heating cycle can be programmed from 0.3125 to 320 ^oC/min - C_p can be obtained, Figure 4-14

- for amorphous polymers, C_p x T_g ≈ 115 J/g - crystallinity,

(4.25) -

- - H

f

Q



 



heat of fusion of semicrystalline polymer

heat of fusion of 100% crystalline polymer

  ^- ^- ^- ^(4.26)

1

p am p

C C



 

 

of semicrystalline polymer at T_g of amorphous polymer at T_g

overestimates  when crystallinity is low because C_p may be

depressed(more decreased) by dispersed small disordered crystallites 0(100% crystalline) ~ (C_p)_am(100% amorphous)

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T_g

excess cold crystallization (exothermic)

T_m

crystalline melting (endothermic) endotherm

slope = C_p

(midpoint)

semicrystalline at RT

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DTA (differential thermal analysis)

- the same heat is provided to both sample and reference; measures T difference (T) - glass transition: T is big near Tg because of specific heat difference

- T_m: absorption peak near T_m due to heat of fusion (latent heat) during heating

drop due to C_p increase at T_g

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Heat-Distortion Temperature

• application-oriented measure of a thermal transition temperature (T_g or T_m)

• heat-distortion (heat-deflection) temperature (HDT)

• ASTM D648:

- T at which a sample bar (127x13x3 mm) deflects by 0.25 mm under a standard load of 455 kPa placed at its center

- heating rate = 2 ^oC/min

• for amorphous polymer, HDT is slightly (10~20 ^oC) lower than T_g

• for semicrystalline polymer, HDT is more closely identified with T_m, Table 4-7

• HDT is upper limit for structural application of the polymer

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- give a sinusoidal strain input; measure stress-strain output of a polymer - free vibration: give a big strain at time=0, and then trace vibrating procedure

ex) torsion-pendulum (Figure 5-2)

- compulsory vibration: give a sinusoidal strain continually, and then measure stress-strain behavior

ex) DMA(dynamic mechanical analysis) Fig.10-17

Dynamic Mechanical Analysis

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stored

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shear deformation:

G^* = G′ + iG″, Tan δ = G″/G′

G^*: complex modulus G′: storage modulus G″: loss modulus

G′

https://en.wikipedia.org/wiki/

Dynamic_mechanical_analysis

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• dielectric constant (= relative permittivity): the ratio between the

capacitance of a capacitor when vacuum is applied between two parallel electrodes and the capacitance of the capacitor when a sample is placed between two electrodes; insulation characteristics ↓ as the ratio ↑

• dielectric constant of a polymer with polar groups:

ε = 4πlC/A

l : sample thickness A : area of the sample

C : capacitance of the capacitor

• how to measure dielectric properties of a sample:

- place a sample between two parallel electrodes of a capacitor

→ apply AC electric field of a certain frequency

→ measure polar group response

complex dielectric constant, ε^* = ε′ - iε″, tan δ = ε″/ε′

- experiments at various frequencies, Fig. 10-18

Dielectric Analysis

off on

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frequency relaxation

temp. = constant

relax. time < exp. time relax. time > exp. time (enough time to relax)

ref) 김성철 외, 고분자공학I, 희중당, 1994

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ε′

temp.

tan d

frequency = constant

• dielectric property change with temp. at a constant frequency:

T_g

relax. time > exp. time relax. time < exp. time (enough time to relax)

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 NMR: apply a magnetic field to a polymer sample with atoms with

magnetic moment like H; see peak change; determine T_g; Fig. 10-20 - below T_g: limited interactions by fixed hydrogens only; wide peak

- above T_g: whole hydrogens respond due to mobility increase; sharp peak