lecture 12 kinetic treatment of the glass transition.ppt · enthalpy changes in the glass...

Relaxation in Glass

Lecture 12: The Glass Transition as a KineticLecture 12: The Glass Transition as a Kinetic Transition

Enthalpy Changes in the Glass Transition Range

H(T) decreases continuously with cooling supercooled

Slope of the H(T) curve is the heat capacity which changes from liquid-like to solid-like lp

y

pliquid

Cpliquidglassvalues in the transition region

Change in heat capacity at the glass transition Cp(Tg) ar

ent

hal

liquid

Hmelting

gfast

Cpglassmeasures the differences between the liquid and solid (glassy) Cp values

Mol

a

slowmelting

crystal Cpcrystal

pglass

Sub-Tg annealing and relaxation can occur if liquid is given sufficient time to relax to l th l t t

Temperature

crystal pcrystal

[email protected] Lecture 12: Thermodynamics of the Glass Transition 2

lower enthalpy state

Entropy Changes in the Glass Transition Range

supercooledliquid

opy glassfast Cpliquid/T

lar e

ntro liquid

ast

Smelting

Mo

crystalslow

Cpcrystal/T

Temperature

crystal

Tm


Gibb’s Free-Energy Changes in the Glass Transition Range

G = H - TS Gibbs’ Free-Energy change at Tm

Liquid

is continuous, there is no “Latent Free-Energy Change” as is the case for the enthalpy and entropy En

ergy Crystal

At the melting pointGliquid = Gcrystal

Below the melting point b’s F

ree-

E

Gliquid > Gcrystal andGcrystallization < O

Above the melting pointG

ibb

g pGliquid > Gcrsytal

Gmelting < O At any point

)()( TSTTG

P

Temperature Tm


At any point T P

Gibb’s Free-Energy Changes in the Glass Transition Range

Glasses then “fall off” the liquid line at progressively lower

Liquid

temperatures the slower the cooling rate

Gibbs’ Free-Energy of the glass ergy Crystal

behaves more like the crystal than the liquid

Glass transition range is the Free

-En

Glasses

range of T where the Gibb’s Free-Energy changes from “liquid-like” values to “solid-like”

lG

ibb’

s Fvalues


Temperature Tm

Fundamentals of the Glass Transition

The Glass Transition is a Kinetic Transition Continuous changes in structure and properties Structure and properties are continuous with

temperatureS d i b h d Structures and properties can be changed continuously by changing the kinetics of the cooled or reheated liquidor reheated liquid


Time and Temperature Dependence of Properties

TEnthalpy (heat content) of sample

At high temperatures, the liquid can reach

H

<< t T or

Property (volume) of sampleliquid

liquid can reach equilibrium after T step, relaxation time is short At low

h

erty

P o

r H

~ tcompared to time allowed

temperatures, the liquid cannot reach equilibrium after T step is long

Pro

pe

Sample T

T step, is long compared to time allowed

>> t

glass

Average cooling rate,

T = T/ t•


time t

Temperature dependence of the internal time scale

While the external time scale, t most often does not change,

Rearrangement of the liquid requires breaking of bonds between atoms (ions)

tTT

/ The internal timescale can be

strongly temperature

( ) This requires thermal energy The relative magnitude of the

energy barrier to motion, Eact strongly temperature dependent, to the available thermal energy,

kT determines the probability of “getting over” the energy barrier

Eact

kTET act

o exp)(

Arrhenius temperature dependence of the “relaxation time”


time

Temperature dependence of the internal relaxation time

For Eact > 0

kTET act

o

exp)(

E

0 1

kTET actexp/)( 0

1.E+181.E+20

Temperature dependence of tau

1000

It is a thermal “probability” of motion

1 E+101.E+121.E+141.E+161.E 18

/tau 0

10000

50000

100000

200000

High T, kT ~ Eact, high probability of motion 1.E+04

1.E+061.E+081.E+10

tau(

T)/

Low T, kT << Eact low probability of motion

1.E+001.E+02

0 500 1000 1500 2000T(K)


T(K)

Temperature dependence of the internal relaxation time

For Eact > 0kT

ET act

o 303.2)(log10

0 1

It is a thermal “probability” of 1.0E+20

Arrhenius Temperature dependence of tau

1000p ymotion

High T, kT ~ Eact, high 1 0E+12

1.0E+14

1.0E+16

1.0E+18

u)

1000

10000

50000

100000

probability of motion

Low T, kT << Eact low b bilit f ti

1.0E+06

1.0E+08

1.0E+10

1.0E+12

tau/

tau 0

probability of motion

1.0E+00

1.0E+02

1.0E+04

0 0.001 0.002 0.003 0.004

1/T(K)


Glass Transition is a Kinetic Transition

Glass formation is a kinetic transition, therefore, it depends upon the kinetics of the process The internal timescale, , for the process is controlled by the , , p y

atomic or ionic bonding between atoms or ions Strong and numerous bonding increases the viscosity Weak and limited bonding decreases the viscosity Weak and limited bonding decreases the viscosity Viscosity relaxation time, = G

The external timescale, t, is controlled by the experiment or process i e how fast is the liquid cooledprocess, i.e., how fast is the liquid cooled Is it purposefully quenched very fast? t is short Is it just allowed to cool naturally under prevailing conditions? Or is it “insulated” and allowed to cool very slowly, t is long



Assume that the “relaxation” following a temperature jump is also “exponential”

Note that ~ 5 relaxation time are required for nearly

)(exp)(

Ttt

Note that 5 relaxation time are required for nearly complete relaxation

%1~010~00705exp5exp

1.0E+18

1.0E+20

Arrhenius Temperature dependence of tau

1000

10000

Note that 99% of change

%101.0007.05expexp

1.0E+08

1.0E+10

1.0E+12

1.0E+14

1.0E+16

1.0E 18

au/ta

u)

10000

50000

100000

has occurred Now consider

E 50 000 J/mol 10-13 sec1.0E+00

1.0E+02

1.0E+04

1.0E+06

1.0E 08

0 0.001 0.002 0.003 0.004ta

Eact ~ 50,000 J/mol, 0 ~ 10-13 sec


1/T(K)

Glass Transition is a Kinetic TransitionNo consider an e ample Now consider an example:

Consider starting at a temperature, T0 above Tm where the viscosity is low and the relaxation time is shortlow and the relaxation time is short compared to an experimental time step t following a temperature step T: m

e

supercooledliquid

liquid

))(1()(),(),(

0000

000

TTVTTVVttTVVtTV

T:

lar V

olum liquid

Vfusion

glass

N f th d t

Mol

crystal crystal

Now, for the second step suppose the relaxation time is still short compared to an experimental time step t following a temperature step Temperaturestep t following a temperature step T:

TTTTVttTV )(1))((1()2,( 00


nTTVtntTV ))(1(),(... 00


Now consider the case where the extent of relaxation depends upon the time it takes for relaxation to supercooledthe time it takes for relaxation to occur:

ume

li id

supercooledliquid

lliquid

)( tt

The amount of relaxation:

olar

Vol

u liquidVfusion

glass

)(

exp)(T

t

0 1 ( t/0) 1 l t TemperatureM

o

crystal crystal

)(exp1)(1

Ttt

0 1 - (-t/0) = 1 complete change

1 - (-t/) = 0, no change

Temperature

( ) , g



Relaxation, therefore depends upon the and t relationship:

)(exp1)(1

)(exp1)(),(

)0,(

1110

1110011

00

TtTTV

TtTTVVtTV

VTV

)()( 11 TT

The original volume

The instantaneous volume change due to T1 change to T1

Th t t f l ti ft t ti t t T ith l ti ti (T )The extent of relaxation after t1 time step at T1 with relaxation time (T1)


Temperature and Time Dependent Volume

After two time steps…

),( 000 VtTV

)(exp1)(1)0,(

11101 T

tTTVtTV

)(exp1)(1),(

1101 T

tTTVtTV

)(exp1))(),(),()2,(

222112

t

TtTTtTVtTVtTV

1))(1)(1

)(exp1)(1)2,(

1102

tTTtTTV

TtTTVtTV


)(

exp1))()(

exp1)(12

221

10 TTT

TTTV

Exercises..…

t = 1 second 0 = 10-13 seconds

For Eact = 50,000 cal/mole (300K) ?

Eact = 5,000 cal/mole (300K) ?

Not relaxedFor E = 50 000 cal/mole

Fully relaxed (80K)?

For Eact = 50,000 cal/mole (1000K)? (80K)?

not relaxed

Relaxed

E (60K)?

KllRRTET act

o

/9871

exp)(


KmolecalR /987.1

Exercises..…

Now consider the “elastic” volume changeT k V 35 l/ l

What is V at 800K?

Take V0 ~ 35 ml/mol Take liquid ~ 400 ppm/K Take T = 1 K

What is at 800K?

What is Tg? Take T = 1 K Start at 1200 K What is V at 1170K?

What is ~ Tg?

What is at 1170K?

nTTVtntTV ))(1()(


TTVtntTV ))(1(),( 00

Exercises…

Now add relaxation component…

Take V ~ 35 ml/mol

What about at 1170K? What about 1100K?

Take V0 ~ 35 ml/mol Take liquid ~ 400 ppm/K Take T = 1 K Take t = 1 second Start at 1200 K

)(exp)(

Ttt

What is V at 1199K, 1 second? RT

ET acto

exp)(

What is at 1198K, 2 seconds?

KmolecalR /987.1

exp1))(1(),( 111011

tTTVtTV


)(

exp1))(1(),(1

11011 TTTVtTV

The Glass Transition from Arrhenius T dependence of taud i l l iand exponential relaxation

Volume vs. Temperature

34

10oC/sec

-1000oC/sec

32(ml/m

ol) -10oC/sec

-1oC/sec32

V(

tau = 1 x 10-13secs exp(50 000cal/mol/RT)- "0" oC/sec

30400 500 600 700 800 900 1000

Temperature (K)

tau = 1 x 10 13secs exp(50,000cal/mol/RT)


Temperature (K)

Homework…

Reconstruct Glass Transition cooling behavior using -10 C/second cooling rate, Arrhenius temperature dependence of relaxation time and exponentialdependence of relaxation time and exponential relaxation function.

Extra Credit… Reconstruct faster (1000 C/sec) and slower cool (1

C/sec) cooling ratesE t E t C dit Extra Extra Credit…

Construct reheating curve for -10 C/second cooling curve at a reheat rate of + 10C/secondcurve at a reheat rate of 10C/second

Extra Extra Extra Credit, Construct reheating curve for -1 and -1000 C/second cooling curves at a reheat rate of + 10C/ d10C/second


lecture 12 kinetic treatment of the glass transition.ppt · enthalpy changes in the glass...

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