Federal

University

of Bahia

INSTITUTE OF HUMANITIES, ARTS

& SCIENCES ‘MILTON SANTOS’

mlfn@ufba.br

Vladimir Mihailovich Fokin

Marcio Luis Ferreira Nascimento

Edgar Dutra Zanotto

Vavilov State Optical Institute, St. Petersburg, Russia

Federal University of São Carlos, Brazil

Federal University of Bahia, Brazil

VITREOUS MATERIALS LAB

SÃO CARLOS - BRAZIL

Dynamic Processes in a Glass-

forming Liquid from very Low

to Deep Undercoolings

Federal

University

of Bahia • MOTIVATION and OBJECTIVE

• STRATEGY and METHODS

• DATA DIGGING and ANALYSES – Viscosity: data

– Crystal growth rates: models and data

– Nucleation time-lags: experiments and data

– Ionic Conductivity: experiments and data

• RESULTS – Diffusion coefficients calculated from:

crystal growth rates U , nucleation time-lags , viscosity , conductivity , direct self-diffusion

measurements (for Li, O and Si) and calculated effective diffusion coefficients.

• CONCLUSIONS

OUTLINE

Federal

University

of Bahia

• The diffusion processes controlling crystal nucleation and growth in complex glass forming liquids (e.g. oxides) have been a subject of intense debate and controversy but are still unknown. For example:

Does the Stokes-Einstein or Eyring (SE) equation breakdown? i.e. is there a decoupling between D calculated by the SE equation and D at:

Which moving units control crystallization? Single atoms or is it a cooperative movement of “molecules”?

MOTIVATION1

log10 D

1/T

D D

Tg

Td = 1.1-1.2Tg?

Federal

University

of Bahia

Crystallization theories typically contain a transport

and a thermodynamic term:

I(T) = (K/3) DI exp(W*/kBT)

U(T) = (K´/) DU [1exp(G/kBT]

transport thermodynamic

Most authors use viscosity data and the SE / E

equation to estimate D

TkDDD B

IU

MOTIVATION2

Federal

University

of Bahia

• Our objective is to shed light into the previous questions by comparing 6 types of diffusion coefficients in Li2O2SiO2 glass:

• calculated from crystal growth rates, DU from nucleation time-lag, D, from viscosity, D, conductivity D, and calculated effective Deff’s;

• with directly measured self-diffusion coefficients, Dcation.

OBJECTIVE

(*) from Zanotto‘s thesis

Federal

University

of Bahia

Crystalline Silica (Quartz, Sand) Silica Glass

Silicon

Oxygen

GLASS DEFINITION1

M. L. F. Nascimento. J. Mat. Educ. 37 (2015) 137-154

Federal

University

of Bahia

Glass: non-crystalline solid that presents

glass transition phenomenon V

olu

me

T melt

T g Temp.

Glass transition

temperature

definition Tg:

volume variation

with temperature.

GLASS DEFINITION2

M. L. F.

Nascimento. J.

Mat. Educ. 37

(2015) 137-154

Federal

University

of Bahia

STRATEGY

We measured, collected and analyzed extensive literature data on crystal growth rates, nucleation, time-lag, viscosity and self-diffusion coefficients in a wide temperature range - between the glass transition and the melting point - of “stoichiometric” glasses that:

i) nucleate in the volume:

Lithium disilicate: Li2O2SiO2

ii) only nucleate at surface:

Silica: SiO2

Federal

University

of Bahia

Crystal growth, viscous flow, conductivity, nucleation time-lag and

self-diffusion coefficients, plus effective diffusion coefficient (the

last is theoretical...)

1. Data Digging &

Analysis of 6 Kinetic

Processes

Federal

University

of Bahia

VISCOSITY DATA

ANALYSIS

Federal

University

of Bahia

TkD B

George Stokes Albert Einstein Henry Eyring

Gustav Tammann Gordon Fulcher

0

10TT

BA

logSvante Arrhenius

T

BA10log

DIFFUSION

VISCOSITY

Federal

University

of Bahia

600 800 1000 1200 1400 1600 1800

0

2

4

6

8

10

12

14400 600 800 1000 1200 1400

0

2

4

6

8

10

12

14

Bockris et al.

Fokin et al.

Gonzalez-Oliver

Heslin & Shelby

Marcheschi

Matusita & Tashiro

Ota et al.

Shartsis et al.

Vasiliev & Lisenenkov

Wright & Shelby

Zanotto

Zengh

log

10

(P

a·s

)

T (K)

Heslin & Shelby*

Izumitami

Joseph

Matusita & Tashiro*

T (oC)VISCOSITY

05491833886234210 ./..log T

James Shelby

Carlos

Gonzalez-Oliver

John Bockris

Rikuo Ota

Li2O2SiO2

Federal

University

of Bahia

800 1000 1200 1400 1600 1800

0

2

4

6

8

10

12

14

Experimental data

VFTH: log10

A B/(T T0)

A 2,66234; B 3432,53748; T0 490,70698) :

2 = 0,02515

Mauro: log10

A (12 A)[Tg/T]exp[(m/(12 A)) 1][(T

g/T) 1]

A 1,21646; m 43,58019; Tg = 724,44471:

2 0,02384

log

10

(P

a·s

)

T (K)

COMPARISON

VISCOSITY MODELS

VFTH, Dienes-Macedo-

Litovitz and Mauro’s

proposals are very similar

Federal

University

of Bahia

CRYSTAL GROWTH

DATA ANALYSIS

(The best fittings are shown by a red line)

Federal

University

of Bahia

i) Normal (N)

ii) Screw Dislocations (SD)

iii) Surface Nucleation (2D)

CLASSICAL CRYSTAL

GROWTH MODELS1

Federal

University

of Bahia

RT

GDTU U exp1

RT

GDfTU U exp1

GT

ZDCTU U exp

2

m

m

m T

TT

V

Gf

24

B

m

k

VZ

3

2

TGNCC S ,,,

i) Normal (N)

ii) Screw Dislocations (SD)

iii) Surface Nucleation (2D)

Only one adjustable parameter for all: l 2D: unknown surface energy

TkDD B

U

CLASSICAL CRYSTAL

GROWTH MODELS2

Federal

University

of Bahia

700 800 900 1000 1100 1200 1300

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

400 500 600 700 800 900 1000

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

U (

m/s

)

T (K)

This work

Barker et al.

Burgner & Weinberg

James

Matusita & Tashiro

Ota et al.

Schmidt & Frischat

Zanotto & Leite

Fokin

Soares Jr.

Gonzalez-Oliver et al.

Deubener et al.

Ogura et al.

T (oC)

GROWTH

Michael Weinberg

Joachim Deubener

Peter James

RT

GDfTU U exp1

SD growth = 0.35Å

Tg = 454oC

1.1Tg

Li2O2SiO2

Federal

University

of Bahia

TIME LAG DATA

Federal

University

of Bahia

indt3

Fokin, Zanotto,

Yuritsyn & Schmelzer

352 (2006) 2681

• “Model” glass that shows nucleation in

volume.

Li2O2SiO2

Fixed Temperature NV

time

tangent = NV / t = I

tind

I = nucleation rate [m3s1]

40min at 455oC + 14min at 620oC

20m

1 ind

2

2

2

indind

12

6 m

m

V

t

tm

mt

t

tI

tNexp

Collins-Kashchiev

NUCLEATION & TIME LAG1

Federal

University

of Bahia

Classical Nucleation Theory has many problems about the pre-exponential factor N0, diffusion mechanisms

(DI), the dependence of surface energy =(r,T), metastable phases (GV) etc...

Tk

WDNI

B

I

*

exp20 2

3

3

16

VGW

*

Iwan Stranski Rostislav

Kaischew

Yakov Zeldovich David

Turnbull

Josiah W. Gibbs Max Volmer Richard

Becker

• In this work we will simple assume that diffusion for crystal growth and nucleation are near the same. We fixed N0

and as feasible parameters.

NUCLEATION & TIME LAG2

Federal

University

of Bahia TIME LAG

Vladimir Fokin

Li2O2SiO2

T/.ln 2876360279.185

1.30x10-3

1.35x10-3

1.40x10-3

1.45x10-3

6

8

10

12

14760 750 740 730 720 710 700 690

6

8

10

12

14

ln

(s)

1/T (K1

)

Fokin et al.

James

Zanotto

Tuzzeo

T (K)

Peter James

Federal

University

of Bahia

700 800 900 1000 1100 1200 1300

107

108

109

700 800 900 1000 1100 1200 1300

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

Tg

I (m

3s

1)

T (K)

Fokin

James

Tuzzeo

Zanotto

Zeng

Tm

U (

m/s

)

T (K)

This work

Barker et al.

Burgner & Weinberg

Deubener et al

Fokin

Gonzalez-Oliver et al.

James

Matusita & Tashiro

Ogura et al.

Ota et al.

Schmidt & Frischat

Soares Jr.

Zanotto & Leite

NUCLEATION & GROWTH

RT

GDfTU U exp1

SD growth = 0.35Å

1.1Tg

Li2O2SiO2

Tk

WDNI

B

I

*

exp20

Federal

University

of Bahia

CONDUCTIVITY DATA

ANALYSIS

Federal

University

of Bahia

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

3.5x10-3

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

120001500 1000 500

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

log

10

(

1cm

1)

1/T (K1

)

Bockris et al.

Dale et al.

Hahnert et al.

Higby & Shelby

Kone et al.

Konstanyan & Erznkyan

Leko

Mazurin & Borisovskii

Mazurin & Tsekhomskii

Pronkin

Souquet et al.

Vakhrameev

Yoshiyagawa & Tomozawa

Tg

T (K)

CONDUCTIVITY

Jean Louis Souquet Ana Candida Rodrigues

Li2O2SiO2

John Bockris

Federal

University

of Bahia

RTGf

UDU

/exp

1

D from

growth:

D from viscous flow:

TkD B

2

LieN

TkD B

(for normal or screw

dislocation growth)

D from time- lag:

22

2

3

80

G

TVkD mB

SIX DIFFERENT DIFFUSION

COEFFICIENTS

Dcation : measured self-diffusion coefficients

D ‘effective’:

Si

Si

O

O

Li

Li

11

D

x

D

x

D

x

D

xD

i

ieffective

D from

conductivity: Nernst-Einstein

Equation

Eyring Equation

Federal

University

of Bahia

2. Results for

Li2O2SiO2

M. L. F. Nascimento, V. M. Fokin, E. D. Zanotto, A. S. Abyzov. Dynamic processes in a

silicate liquid from above melting to below the glass transition. J. Chem. Phys. 135 (2011)

194703

Federal

University

of Bahia DIFFUSIVITY1

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

1.6x10-3

1.8x10-3

2.0x10-3

-22

-20

-18

-16

-14

-12

-10

-82000 1500 1000 500

D

DU

log

10 D

(m

2/s

)

1/T (K1

)

Tm

Tg

T (K)

ONLY DU

Li2O2SiO2

Federal

University

of Bahia

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

1.6x10-3

1.8x10-3

2.0x10-3

-22

-20

-18

-16

-14

-12

-10

-82000 1500 1000 500

DU

D=k

BT/

log

10 D

(m

2/s

)

1/T (K1

)

Tm

Tg

T (K)

Breakdown

at 1.1Tg ?

DU & D

DU D for T > 1.1Tg

D < DU low T

DIFFUSIVITY2 Li2O2SiO2

Federal

University

of Bahia

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

1.6x10-3

1.8x10-3

2.0x10-3

-22

-20

-18

-16

-14

-12

-10

-82000 1500 1000 500

DO: Takizawa et al.

D*

DO: Sakai et al.

DU

D=k

BT/

DLi

: Beier & Frischat

DO: Takizawa et al.

DO: Sakai et al.

D*: Kawakami et al.lo

g1

0 D

(m

2/s

)

1/T (K1

)

Tm

Tg

DLi

T (K)

DU, D & D’s

DU D D* for T > 1.1Tg

DLi > DO > DU > D low T

DIFFUSIVITY3 Li2O2SiO2

Federal

University

of Bahia

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

1.6x10-3

1.8x10-3

2.0x10-3

-22

-20

-18

-16

-14

-12

-10

-82000 1500 1000 500

D

DO: Takizawa et al.

D

D*

DO: Sakai et al.

DU

D=k

BT/

DLi

: Beier & Frischat

DO: Takizawa et al.

DO: Sakai et al.

D*: Kawakami et al.

D

log

10 D

(m

2/s

)

1/T (K1

)

Tm

Tg

DLi

T (K)

DU, D, D’s & D

2ne

TkD Bn = 51028 m3

DIFFUSIVITY4 Li2O2SiO2

Federal

University

of Bahia

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

1.6x10-3

1.8x10-3

2.0x10-3

-22

-20

-18

-16

-14

-12

-10

-82000 1500 1000 500

D

DO: Takizawa et al.

D

D*

DO: Sakai et al.

DU

D=k

BT/

DLi

: Beier & Frischat

DO: Takizawa et al.

DO: Sakai et al.

D*: Kawakami et al.

D

D

log

10 D

(m

2/s

)

1/T (K1

)

Tm

Tg

DLi

T (K)

DU, D, D’s , D & D

223

80

V

B

G

TkD

DIFFUSIVITY5 Li2O2SiO2

Federal

University

of Bahia

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

1.6x10-3

1.8x10-3

2.0x10-3

-22

-20

-18

-16

-14

-12

-10

-82000 1500 1000 500

D

DO: Takizawa et al.

D

D*

DO: Sakai et al.

DU

D=k

BT/

DLi

: Beier & Frischat

DO: Takizawa et al.

DO: Sakai et al.

D*: Kawakami et al.

D

D

DI

log

10 D

(m

2/s

)

1/T (K1

)

Tm

Tg

DLi

T (K)

Tk

W

N

ID

B

I

*

exp0

2

= 0.1584 J/m2

N0 = 9.781027 m3

DU, D, D’s , D , D & DI

DIFFUSIVITY6 Li2O2SiO2

Federal

University

of Bahia

1.2x10-3

1.3x10-3

1.4x10-3

1.5x10-3

1.6x10-3

1.7x10-3

-23

-22

-21

-20

-19

-18

-171.2x10

-31.3x10

-31.4x10

-31.5x10

-31.6x10

-31.7x10

-3

-23

-22

-21

-20

-19

-18

-17

log

10 D

(m

2/s

)

1/T (K1

)

DU

DI

DO: Takizawa et al.

DO: Sakai et al.

Tg = 454

oC

DO

1.1Tg

1/T (K1

)

D

D

log10 DU = 5.5508 + exp(352.4 kJ/RT)

log10 D = 18.8486 + exp(559.5 kJ/RT)

DU D DI >> D

In the nucleation range

the viscosity strongly

decouples from crystal

growth, nucleation and

time lag experimental

data. Oxygen diffusion

does not follow D but

DU.

DIFFUSIVITY7 Li2O2SiO2

Federal

University

of Bahia

COMPARISON WITH

SIMULATION: MD

4,0x10-4

5,0x10-4

6,0x10-4

7,0x10-4

8,0x10-4

9,0x10-4

1,0x10-3

-13

-12

-11

-10

-9

-8

log

10 D

(m

2/s

)

1/T (K1

)

D=k

BT/ : 0.3Å

DLi

: MD

DSi

: MD

DO: MD

Deff

D=k

BT/ : 2.7Å

José Pedro Rino

L. G. V. Gonçalves, J. P.

Rino. J. Non-Cryst.

Solids 402 (2014) 91-95

Federal

University

of Bahia

Crystal growth, viscous flow, silicon and oxygen self-diffusion in a

silicate glass that does not display nucleation in volume at

laboratory time-scales

3. Results for SiO2

M. L. F. Nascimento, E. D. Zanotto. Diffusion Processes in Vitreous Silica Revisited. Phys.

Chem. Glasses 48 (2007) 201-216

Federal

University

of Bahia

= 2Å

Normal growth

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

9.0x10-4

-24

-23

-22

-21

-20

-19

-18

-17

-16

-152200 2000 1800 1600 1400 1200

log

10 D

(m

2/s

)

1/T (K1

)

DU: Wagstaff

Tg = 1451 K

Tm = 2007 K

T (K)

DIFFUSIVITY1 SiO2

Federal

University

of Bahia

DU D at T > 1.1Tg

= 2Å

Normal growth

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

9.0x10-4

-24

-23

-22

-21

-20

-19

-18

-17

-16

-152200 2000 1800 1600 1400 1200

log

10 D

(m

2/s

)

1/T (K1

)

DU: Wagstaff

D: Brebec et al.

Tg = 1451 K

Tm = 2007 K

T (K)

DIFFUSIVITY2 SiO2

Federal

University

of Bahia

DU D at T > 1.1Tg

DSi Dh at T < 1.1Tg

= 2Å

No breakdown

with U till 1.1Tg

and with DSi T < Tg

Normal growth

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

9.0x10-4

-24

-23

-22

-21

-20

-19

-18

-17

-16

-152200 2000 1800 1600 1400 1200

log

10 D

(m

2/s

)

1/T (K1

)

DU: Wagstaff

D: Brebec et al.

DSi

: Brebec et al.

Tg = 1451 K

Tm = 2007 K

T (K)

DIFFUSIVITY3 SiO2

Federal

University

of Bahia

DU D at T > 1.1Tg

DSi Dh at T < 1.1Tg

M. L. F. Nascimento, E. D.

Zanotto Phys. Rev. B 73 (2006)

= 2Å

No breakdown

with U till 1.1Tg

and with DSi T < Tg

Normal growth

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

9.0x10-4

-24

-23

-22

-21

-20

-19

-18

-17

-16

-152200 2000 1800 1600 1400 1200

log

10 D

(m

2/s

)

1/T (K1

)

DU: Wagstaff

D: Brebec et al.

DSi

: Brebec et al.

DO: Kalen et al.

Tg = 1451 K

Tm = 2007 K

T (K)

DIFFUSIVITY4 SiO2

Federal

University

of Bahia

4. Conclusions

Federal

University

of Bahia

log10 D

1/T

D D

Tg

Td ~ 1.1-1.2Tg?

i) Does the SE/ E equation breakdown at some low

enough T?

INITIAL QUESTIONS

ii) Which moving units control crystallization? Single

atoms or is it a cooperative movement of molecules

(Deff)?

Federal

University

of Bahia

DO

DLi

SKETCH

D

log10 D

1/T

Td ~ 1.2-1.1Tg

Tg

DSi

DO?

Federal

University

of Bahia

DO

DSi

DLi

log10 D

1/T

Td ~ 1.2-1.1Tg?

Time lag

Tg

DO?

Deffective D

( = 2.7Å)

D

CONCLUSION: SKETCH

DU (surface) > Deff (volume)

Federal

University

of Bahia

Which ions control I and U in oxide glasses?

As expected, at low T the alkalis diffuse much faster than silicon, oxygen and whatever “molecules” control viscous flow, crystal nucleation and growth. However, near Tm the diffusivities of all ions are similar.

For SiO2 glass, silicon diffusivity controls crystal growth in the whole T range! There is no data for the other glass studied here…

Is there a breakdown for SE/E or not?

DU showed departures from D starting at T1.1Tg for some systems, but there are few exceptions (not shown). Therefore, these departures for some systems could be a sign of a possible breakdown of the SE / E equation.

CONCLUSIONS1

Federal

University

of Bahia

For silica glass there is a remarkable decoupling of (possible non-bridging) oxygen with D, but not silicon above Tg!!!

The temperature dependence of diffusivity calculated from time-lag (D) do not agree with D even below 1.1Tg.

The diffusivities calculated from the Nernst-Einstein

relationship for ionic conductivity (considering fixed the

number of diffusing ions ~1028m3) agree with directly

measured diffusivity data.

The VFTH equation fits well the viscosity data of most

authors for all glasses from Tm to Tg!

This study validates the use of viscosity (through the SE/E equation) to account for the kinetic term of the crystal growth expression in a wide range of temperatures above 1.1Tg.

CONCLUSIONS2

Federal

University

of Bahia

Acknowledgments

THANK YOU!!!!

FROM LAMAV BASIS AT SALVADOR, BAHIA

Bo

ipeb

a

Mar

aú

Sal

vad

or

Federal

University

of Bahia

Dedução das Leis

de Fick

Federal

University

of Bahia

A. E. Fick, Ann. Phys. 170 (1855) 59 A. E. Fick, Phil. Mag. 10 (1855) 30

Adolf Eugen Fick (1829-1901), médico e fisiologista alemão

Federal

University

of Bahia

Difusão atômica: transporte de átomos ou moléculas

Suponha um gás G1 numa caixa em equilíbrio térmico e introduzindo uma pequena quantidade de um outro gás G2

dentro desta caixa, este gás se espalha aos poucos devido às colisões que sofre entre suas partículas e entre o gás G1. Este processo é chamado de DIFUSÃO.

Considere o fluxo de gás na direção x e um plano A perpendicular a x :

Difusão em Sólidos1

n n

vt x

área A

vt

Federal

University

of Bahia

Considere ainda o número que atravessa uma superfície A em um tempo t dado pelo número de partículas que estão a uma distância vt de A. (onde v é a velocidade molecular real )

número de partículas que atravessam a superfície da esquerda para direita:

número de partículas que atravessam da direita para a esquerda:

Difusão em

Sólidos2

Para tanto, é necessário calcular o fluxo total (J) e considerar como fluxo positivo aquelas partículas que cruzam na direção positiva de x e

subtraímos o número das que cruzam a mesma superfície na direção negativa de x.

é possível contar o número de

partículas que atravessam A :

n n

vt x

área A

vt

nvt

número de partículas por unidade de volume à esquerda do plano A. n

nvt

número de partículas por unidade de volume à direita do plano A. n

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University

of Bahia

Assim pode-se entender n(x, y , z) como a

densidade de partículas em um pequeno elemento de volume xyz centrado em (x , y , z). Em termos de n podemos expressar

a diferença:

Substituindo: 2

v2 l

dx

dnJ

ldx

dnx

dx

dnnn 2

(onde l é o livre caminho médio)

Definindo J como uma corrente molecular,

ou a densidade de corrente que atravessa o

plano A, então J corresponde ao fluxo total

de partículas por unidade de área por

unidade de tempo:

2

v

2

vv

nn

t

tntnJ

Considere ainda uma distribuição espacial das n partículas por uma função contínua de x, y e z .

Difusão em

Sólidos3 n

n

vt x

área A

vt

dx

dnlJ x vou

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University

of Bahia

Se se substitui l v (onde é o tempo entre colisões) e m (onde é a

mobilidade, e m a massa da partícula) na PRIMEIRA LEI DE FICK obtém-se:

dx

dnmJ x 2v

3

1

mas pelo Princípio

da Equipartição da

Energia tem-se que:

Tkm B2

3v

2

1 2

dx

dnTkJ Bx

Em três dimensões o resultado acima diferencia

de um fator 1/3, devido à isotropia do espaço.

Logo uma resposta melhor é:

dx

dnlJ x

3

v

ou

dx

dnDJ x

PR

IME

IRA

LE

I DE

FIC

K Difusão em

Sólidos4 n

n

vt x

área A

vt

Adolf Gaston Eugen Fick

(1852-1937), médico

oftalmologista alemão,

sobrinho de Fick e inventor

das lentes de contato: Eine

Contactbrille, Archiv für

Augenheilkunde 18 (1888)

285

Federal

University

of Bahia

Difusão em

Sólidos5

TkD B

E se ainda for considerado um fluxo

n de partículas com carga elétrica, a

Equação da Continuidade diz que, em

relação ao princípio de conservação

da carga:

tJ

No caso em que D não depende da concentração n, obtém-se a SEGUNDA LEI

DE FICK:

2

2

dx

ndD

dt

dn

Sendo:

00,,,, xJzyx

J

dx

dnD

dx

dJ

xt

nx

SE

GU

ND

A L

EI D

E F

ICK

Adolf Eugen Fick

(1829-1901), médico

e fisiologista alemão

Federal

University

of Bahia

Resumo das Leis

de Fick

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University

of Bahia

Primeira Lei de Fick1

O fluxo da impureza na direção x é proporcional ao

gradiente de concentração n nesta direção.

Jx : Fluxo de átomos através da área A [átomos/m2s]

D : coeficiente de difusão

ou difusividade [m2/s]

dx

dnDJ x

n

x

A

dx

dn

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University

of Bahia

Estado estacionário J constante no tempo

Ex: Difusão de átomos de um gás através de uma placa

metálica, com a concentração dos dois lados mantida constante.

J

na nb xa xb

posição x

na

nb

ab

abx

xx

nnD

dx

dnDJ

Primeira Lei de Fick2

Federal

University

of Bahia

Segunda Lei

de Fick

A taxa de variação da concentração com o tempo, é igual ao gradiente do fluxo:

2

2

dx

ndD

dt

dn

dx

dnD

dx

d

dt

dn

Se a difusividade não depende de x:

Esta equação diferencial de segunda ordem só pode ser resolvida se forem fornecidas as condições de fronteira.

Exemplo de difusão: oscilações

próximas de posições de equilíbrio

permitem saltos eventuais e

aleatórios para as vacâncias vizinhas