Universidade Federal da Bahia Faculdade de Medicina da Bahia PET- Medicina Ana Souza Marques
Dynamic Processes in a Glass- forming Liquid from … · Federal University of Bahia INSTITUTE OF...
Transcript of Dynamic Processes in a Glass- forming Liquid from … · Federal University of Bahia INSTITUTE OF...
Federal
University
of Bahia
INSTITUTE OF HUMANITIES, ARTS
& SCIENCES ‘MILTON SANTOS’
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
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
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
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
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
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
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
Federal
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
Federal
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
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
Federal
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