a physico-geometric approach to solid-state reactions by

Nobuyoshi KOGAGraduate School of Education

HIROSHIMA University

A Physico-geometric Approach to Solid-State Reactions by Thermal Analysis

OUTLINE

Theme:How to characterize the overall kinetics of the solid-state reactions from physico-geometric points of view?

Contents:(1) Physico-Geometry of Solid-State Reactions(2) Kinetic Model Functions(3) Kinetic Equations(4) Application of Thermoanalytical Method(5) Sample Controlled Thermal Analysis(6) Concluding Remarks

Surface Nucleation (1)ex. Thermal Decomposition of Solids

Reactant Surface one of the most reactive sites

Reaction initiates by nucleation and growthSurface defects and dislocationsCrystallographic directionReaction atmosphereOthers

Nonisothermal dehydration of α-NiSO4⋅6H2O to tetrahydrates under flowing N2.

Surface Nucleation (2)

□ Relation to Crystallographic Direction

plate-like single crystals with stacking layer structure

Edge surfaceThe most reactive sitesProduct nuclei appear parallel to the most developed surface (010)

Isothermal dehydration of zinc acetate dihydrate to anhydride under flowing N2.


□Influence of Reaction Atmosphere

At a reduced pressure of 1.5 hPa

Under flowing N2

Formation of trihyrate nuclei

Formation of monohydrate nuclei

Isothermal dehydration of copper(II) sulfate pentahydrate


□Structure of Product Nuclei

Isothermal dehydration of copper(II) sulfate pentahydrate to trihydrate.


Classification of Product NucleiClassification Characteristics Example

Fluid-flux Retention of a volatile product at interface condensed as liquid

Dehydration of alumKBr+Cl2→KCl+BrCl

FusionProduct melting facilitates reaction

Decomposition of ammonium dichromate and copper(II) malonate

FunctionalStrain and/or heterogeneous catalytic breakdown of intermediate at reaction interface

Decomposition of copper(II) formate, nickel formate, nickel aquarate and silver malonate

Flux-filigree Two-zone structure of reaction interface

Dehydration of Li2SO4·H2O

Reaction Interface (1)

A zone of locally enhanced reactivity located at reactant/product contactReaction Interface

Reaction proceeds by advancement of reaction interface towards the center of reactant crystal

Crystallographic directionShape of reaction frontStructure of reaction

interfaceIsothermal dehydration of copper(II) acetate monohydrate

Reaction Interface (2)Example (1)

Wavy frontSurface nuclei Growing hemisphericallyoverlap

Isothermal dehydration of lithium sulfate monohydrate to anhydride


Reaction front

A saw-like shape

Product layer

A repeated stacking of dense and coarse aggregates of the product crystallites

Isothermal dehydration of copper(II) sulfate pentahydrate to trihydrate


Reaction front Needle-like product crystallites

Penetration to reactant crystal

Surface nucleation at edge surface of plate-like crystal 2D-shrinkage

Isothermal dehydration of zinc acetate dihydrate


Reaction Interface An optically different layer of 20~50mm thickness

Isothermal dehydration of copper(II) acetate monohydrate

Reaction Interface (6)

A Model of Reaction Interface

What is taking place?

ex. Thermal dehydration• Water mobility within the

crystalline reactant.• Diffusive migration of water to

crystal edge or interface.• Water volatilization at the

crystal edge of interface.• Chemical reaction, if any.• Recrystallization of solid

reactant to product structure.• Water vapor adsorption/

desorption on product solid.• Intranuclear diffusive escape of

water and loss beyond crystal.

Kinetic Model Function(1)A hypothetical transfer from homogeneous to heterogeneous model

Kinetic Model Function(2)Typical kinetic model functions for solid-state reaction

Model Symbol

Nucleation & Growth Am (0.5≤m≤4)

Phase Boundary Controlled Rn (1≤n≤3)

D1 (one D)

D2 (two D)

D3 (three D)

D4 (three D)

Diffusion Controlled

( )tk

fdd1 α

α = ( ) ( ) ktf

g =∫=α

ααα 0

d

( ) ( )[ ]m m1 1 1 1− − − −α αln / ( )[ ]− −ln /1 1α m

( )n n1 1 1− −α / ( )1 1 1− −α / n

12α α2

( )−

−1

1ln α α + 1 −α( ) ln 1−α( )

( )( )

3 1 2 3

2 1 1 1 3−

− −⎡⎣⎢

⎤⎦⎥

α

α

/

/ ( )1 1 1 3 2− −⎡

⎣⎢⎤⎦⎥

α /

( )3

2 1 1 3 1− − −⎡⎣⎢

⎤⎦⎥

α / ( )1 23

1 2 3− − −α α /

Kinetic EquationGeneral Kinetic Equation

( )αα kft=

ddPhysico-geometric Kinetic

Model Function

Temperature Dependence of rate constant

Other functional dependence of reaction rate

⎟⎠⎞

⎜⎝⎛−=

RTEAk exp

( ),.....,,dd PTa

tαα

∝

( ) ( ),.....,,expdd PTaf

RTEA

tααα

⎟⎠⎞

⎜⎝⎛−=

( ) 1,.....,, =PTa αIdealized Kinetic Equation

( )αα fRTEA

t⎟⎠⎞

⎜⎝⎛−= exp

dd

Application of TA

Technique Measuring Property Kinetic Rate Data

DTA Temperature difference (Tp, αp)

TG Mass change (dα/dt, α, T, t)

DSC Heat flow rate (dα/dt, α, T, t)

EGA Evolved gas (dα/dt, α, T, t)

Isothermal Analysis(1)

(1) Measurement of Isothermal mass-loss data

0 50 100 150 200 250 3000

50

100

150

200

250

300

T

/ °C

-4

-3

-2

-1

0

DTG

TG

∆m

/ m

g

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03T

(d∆

m/d

t) / m

g m

in-1

Isothermal Analysis(2)(2) Determination of rate constant k

(b) Integral( )αα kf

t=

dd(a) Differential

( ) ktg =α

0 10 20 30 400.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.02 0.04 0.060

5

10

15

20

25

30

0.0

0.5

1.0

1.5

2.0

D3

R2 A1

D2

(b) Integral

g(α)

t / min-1

D3

D2

R2

f(α)

(dα/dt) / min-1

(a) differential

A1

Isothermal Analysis(3)

(3) Determination of E & A

3.02 3.04 3.06 3.08 3.10 3.12 3.14-10.0

-9.5

-9.0

-8.5

-8.0

-7.5

-7.0

D2:

Ea=106.5±5.4 kJ mol-1

A1: Ea=106.6±5.4 kJ mol-1

ln k

/ s-1

T-1 / kK-1

Arrhenius plot

RTEAk −= lnln

Nonisothermal Analysis(1)

(1) Measurement of Nonisothermal rate data

0 40 80 120 160 200 240100

150

200

250

300

350

400

T

T

/ °C

-4

-3

-2

-1

0

DTG

∆m /

mg

-0.08

-0.06

-0.04

-0.02

0.00

0.02

TG

(d∆

m/d

t) / m

g m

in-1

Nonisothermal Analysis(2)From single TA run2) Integral

1) Differential

At φ=const.φααφ

Tt

tTT

dd

dd

0

=

+=

( )αφα fRTEA

T ⎟⎠⎞

⎜⎝⎛−= exp

dd

( ) RTEA

fT−=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ln

ddln

αφα

( )( )

( )xpR

AETRTEA

fg

T

T φφααα

α

=⎟⎠⎞

⎜⎝⎛−== ∫∫ dexpd

00

RTEx =with

( ) ( ) ( ) ( ) ( )xx

xx

xx

xxpx

π−≅

−−

−= ∫

∞ expexpexp

For example( )

xxxx /2−

=πCoats & Redfern method

1.68 1.72 1.76 1.80

-5

-4

-3

-2

-1

0

R3

A1

ln [(

dα/d

t)/f(a

)] /

min

-1

T-1 / kK-1

(a) Sharp & Wentworth

A2

------------------------f(α) Ea / kJ mol-1

------------------------ A1 234±1A2 122±1R3 184±3------------------------

( )RTE

ERT

RAE

Tg

−⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −=

21lnln2 φα

1.68 1.72 1.76 1.80

-16

-15

-14

-13

-12

R3

A2

A1

(b) Coats & Redfern

ln [g

(α)/T

2 ] / K

-2

T-1 / kK-1

Nonisothermal Analysis(3)From a series of TA runsIsoconversional method

( )αα fRTEA

t ⎟⎠⎞

⎜⎝⎛−= exp

dd 2) Integral

( )( )

( )xpR

AETRTEA

fg

T

T φφααα

α

=⎟⎠⎞

⎜⎝⎛−== ∫∫ dexpd

001) Differential

RTEx =withFriedman method

( )[ ]RTEAf

t−=⎟

⎠⎞

⎜⎝⎛ αα ln

ddln ( )

RTExp 4567.0315.2log −−=Ozawa method

2.20 2.24 2.28 2.32 2.36 2.40 2.44 2.48-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

T-1 / kK-1

α=0.2 α=0.4 α=0.6 α=0.8

ln (d

α/dt

) / m

in-1

(a) Friedman ( ) RTE

RgAE 4567.0315.2loglog −−=α

φ

2.20 2.24 2.28 2.32 2.36 2.40 2.44 2.48-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

log φ

/ K m

in-1

T-1 / kK-1

(b) Ozawa

From a series of TA runs Nonisothermal Analysis(4)Peak method Kissinger method

○ First order equation( )αα f

RTEA

t ⎟⎠⎞

⎜⎝⎛−= exp

dd

f’(α)=-1First derivative with respect to t

△ For other kinetic models( ) ( )

⎥⎦⎤

⎢⎣⎡ +⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−= 22

2

expd

dexpdd

RTE

RTEAff

RTEA

tφ

αααα

At peak maximum

0 20 40 60 80 1000.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

A4

D2

D4

R2R3

A3

A2

α p

xp

A1

0dd

2

2

=⎟⎠

⎞⎜⎝

⎛

ptα

( )p

p

p RTE

EARf

T−⎥⎦

⎤⎢⎣

⎡=αφ '

lnln 2

( ) ( )ααα

dd' ff =with

Nonisothermal Analysis(5)Kinetic Master Plot

Isothermal NonisothermalIntroduction of generalized time θ

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Fractional reaction α

D2

D3

D4

A4

A3

A2

A1

R3

R2f(α)/f

(0.5

)

(a) Differential (b) Integral

g(α

)/g(0

.5)

A4A3

A2

R2

R3

A1D2

D4

D3

( )( )

( )( ) 5.0dd

dd5.0 =

=αα

ααt

tff( )αα kf

t=

dd

( ) ktg =α( )( ) 5.05.0 =

=α

αt

tgg

∫ ⎟⎠⎞

⎜⎝⎛−=

ta t

RTE

0

dexpθ

( )αθα Af=

dd

( ) ( ) θθααα

θα

AAf

g === ∫∫00

dd

( )( )

( )( ) 5.0dd

dd5.0 =

=αθαθαα

ff

( )( ) 5.05.0 =

=αθθα

gg

1) Differential

2) Integral1) Differential

2) Integral

⎟⎠⎞

⎜⎝⎛=

RTE

taexp

dd

dd αθα

( )xpR

Ea

βθ =

Ambiguities of TA Kinetics(1)

Yes!!(1) For simulating the reaction process(2) For estimating life-time(3) Others

□Physico-chemical meaning of the kinetic parameters?

□Practical usefulness of the kinetic parameters?

Not answered as yet!!

Problems(1) Further detailed characterization of reaction mechanism(2) Missing functional dependence in the idealized kinetic equation(3) Reliability of the kinetic rate data measured by TA

1) Self-cooling or self-heating effects2) Self-generated reaction atmosphere3) Mass-transfer and heat-transfer phenomena4) Others

Ambiguities of TA Kinetics(2)Self-cooling effect

0 5 10 15 20 25300

350

400

450

500

550

-1.5

-1.0

-0.5

0.0

-0.4

-0.3

-0.2

-0.1

0.0

0.1

8

9

10

11

12

13

14

T

/ K

t / min

TG /

mg

TTG

β

DTA

/ µV

DTAE

ndo.

β / K

min

-1

Ambiguities of TA Kinetics(3)

Self-generated atmosphere

450 500 550 600 650 700-8

-6

-4

-2

0

2

4

6

8

-30

-20

-10

0

10

20

30

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

TG /

mg

T / K

CO2

Endo

.D

TA / µV

TG

DTA

H2O

Con

cent

ratio

n / g

m-3

Ambiguities of TA kinetics(4)Mass-transfere phenomena

Increase of Sample Mass

Balk or Pellet PowderDifficulty of gross diffusion of product gas through sample power matrix

Difficulty of internal diffusive escape of product gas through solid product layer

1.0

0.8

0.6

0.4

0.2

0.0

Frac

tiona

l rea

ctio

n α

6050403020100

Time / min

2.5 mg 5.0 mg 10.0 mg 20.0 mg

Li2SO4⋅H2O → Li2SO4 + H2O

CRTA(1)

Practical Usefulness of CRTA Techniques(1)Separation of overlapped reaction steps(2)Measurement of kinetic rate data(3)Process control and product control

CRTA(2)

Technique Feedback parameter

CRTG Rate of mass change

CRDL Rate of length change

CREGD Rate of total gas evolution

CREGA Rate of specific gas evolution

CREGD-TG(1)

Thermal Decomposition of Calcite

CaCO3→CaO + CO2( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛−=

0

1expdd

PPf

RTEA

tαα

1.0

0.8

0.6

0.4

0.2

0.0

1-P/

P 0

140012001000800600

T / K

P=10-2

110

102

103

105 Pa

104

P0 in Pa = 1.87x1012exp(-19697/T)

Temperature Dependence of 1-P/P0

CREGD-TG(2)

CREGD-TG(3)

0 50 100 150 200 250 300 350

400

600

800

1000

(d∆W/dt)

∆W

T

T / K

Time / min

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Mas

s cha

nge ∆

W /

mg

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

(d∆

W/d

t) / m

g m

in-1

Typical CREGD-TG Record

CREGD-TG(4)

0.0 0.2 0.4 0.6 0.8 1.0800

850

900

950

1000

1050

1100

0.65 mg 1.14 mg 2.01 mg 3.28 mg 5.43 mg 9.82 mg 15.31 mg 20.70 mg 42.65 mg

T /

K


Effect of Sample MassHow to change decomposition rate for kinetic calculation?

(1)Control pressure

(2)Sample mass

(3)Basal pressure

CREGD-TG(5).1

0

constPP

≈⎟⎟⎠

⎞⎜⎜⎝

⎛− ( )[ ]

RTEAf

t−=⎟

⎠⎞

⎜⎝⎛ αα ln

ddln

1.04 1.08 1.12 1.16 1.20

-11

-10

-9

-8

-7

α=0.80.6 0.4

0.2

0.65 mg 1.14 mg 2.01 mg 3.28 mg 5.43 mg 9.82 mg 15.31 mg 20.70 mg 42.65 mg

ln (d

α/d

t) / s

-1

kK / T

Friedman Plot

CREGD-TG(6)

0 200 400 600 800300

400

500

600

700

800

900

1000

1100T

/ K

Time / min

-20

-16

-12

-8

-4

0

(d∆W/dt)

∆W

T

Mas

s cha

nge ∆W

/ m

g

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

(d∆

W/d

t) / m

g m

in-1

Typical Rate-jump CREGD-TG Record

( )( )2

1

21

21

d/dd/dlntt

TTTTRE

αα

−=

CREGA-TG(1)

CREGA-TG(2)

300 400 500 600 700-4

-3

-2

-1

0

1

Mas

s cha

nge ∆

W /

mg

T / K

-6

-4

-2

0

2

4

6

∆T / µV

TG

DTA

DTG

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

(d∆W

/dt)

/ mg

min

-1

Thermal decomposition of synthetic

malachite TG-DTA

Smooth

Single Stepmass-lose process

E≈190 kJ mol-1

First Order Reaction

Kinetics

Cu2CO3(OH)2 → 2CuO + CO2↑ + H2O↑

CREGA-TG(3)

520 540 560 580 600 620

0.0

0.2

0.4

0.6

0.8

1.0

Static air 60 ml min-1 (N2) 90 ml min-1 (N2) 1.5 Pa

Frac

tiona

l rea

ctio

n α

T / K

0.00

0.02

0.04

0.06

0.08

0.10

0.12

(dα/

dt) /

min

-1

Influence of Reaction Atmosphere With increasing

the influence of self-generated atmospheric

condition

Reaction temperature shifts systematically to lower temperature region

CREGA-TG(4)

700

650

600

550

500

T / K

1.00.80.60.40.20.0Fractional Reaction α

flowing dry N2 flowing dry N2-CO2 flowing wet N2

Dry N2

Dry N2-CO2

Normal

Wet N2

Inverse

Influence of atmospheric CO2 and H2O

CREGA-TG(5)

400 500 600 700 800-10.0

-7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

10.0

-40

-20

0

20

40

60

80

0

1000

2000

3000

4000

5000

0

1

2

3

4

∆T / µV

Mas

s cha

nge ∆W

/ m

g

T / K

∆T

∆W

CO2

H2O

CO

2 / pp

m

H2O

/ g

m-3

Sample mass: 20.08 mg

Carrier Gas: dry N2

Flow Rate: 200 ml min-1

H.R.: 10.0 K min-1

A Typical TG-DTA-EGA Curves

CREGA-TG(6)

0 50 100 150 200300

400

500

600

700

800

-8

-6

-4

-2

0

2

4

6

8

0

100

200

300

0

2

4

6

8

10

∆W

T

T / K

time / min

Mas

s cha

nge ∆W

/ m

g

CO2

H2O

CO2 /

ppm

H2O

/ g

m-3

Typical CREGA(CO2)-TG Curves

Sample mass: 20.55 mg

Carrier Gas: dry N2


H.R.: 2.0 K min-1

Controlled Gas: CO2 (100ppm)

CREGA-TG(7)

500

525

550

575

600

0 -1 -2 -3 -4 -5 -6

H2O= 1.08 g m-3

H2O= 8.16 g m-3

H2O=11.57 g m-3

Mass change ∆W / mg

T / K

Sample mass: 20.55 mgCarrier Gas: N2

with H2O: 1.98 g m-3

8.14 g m-3

11.57 g m-3


H.R.: 2.0 K min-1

Controlled Gas: CO2Controlled Value: 100ppm

Influence of Water Vapor

0.0 0.2 0.4 0.6 0.8 1.0475

500

525

550

575

600

625

0.0 0.2 0.4 0.6 0.8 1.0475

500

525

550

575

600

625(b) H2O = 8.2 g m-3(a) H2O = 1.1 g m-3

CO2= 0.1 g m-3

CO2= 0.3 g m-3

CO2= 0.55 g m-3

T / K


CO2= 0.1 g m-3

CO2= 0.3 g m-3

CO2= 0.55 g m-3

CREGA-TG(8)

Influence of CO2

CREGA-TG(9)

How to change decomposition rate for kinetic calculation?

Parameter Problem

Control Value Change in the partial pressure of evolved gas

Sample Mass Change in the mass-transfer phenomena in the sample matrix

Flow Rate Change in the suction effect

CREGA-TG(10)

0 50 100 150 200 250 300300

400

500

600

700

-8

-6

-4

-2

0

2

4

6

8

0

50

100

150

200

0

1

2

3

4

5

T / K

time / min

Mas

s cha

nge ∆W

/ m

g CO2 /

ppm

H2O

/ g

m-3

Two Different CREGA(CO2)-TG Curves for Kinetic Calculation

Sample Mass: 20.22 mg

Ref. Mass:20.20 mg

Carrier Gas: dry N2


H.R.: 2.0 K min-1

Controlled Gas: CO2 (100ppm)

Conclusion(1)

Application of Dynamic Temperature Change in the Type of Feedback Control

Thermal Decomposition of Solids

Possibility of Quantitative Control of the Relationship between Reaction Kineticsand Self-generated Reaction Atmosphere

Near-equilibrium Far-equilibrium

Conclusion(2)

Logical Consideration on Kinetic Characteristics

Appropriate Control of Reaction Atmosphere and Reaction Rate

Practical Usefulness of CRTA Techniques(1)Separation of overlapped reaction steps(2)Measurement of kinetic rate data(3)Process control and product control

The End

a physico-geometric approach to solid-state reactions by

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