Adolf Eugen Fick

Federal INSTITUTE OF HUMANITIES, ARTS
University & SCIENCES ‘MILTON SANTOS’
of Bahia
VITREOUS MATERIALS LAB
SÃO CARLOS - BRAZIL
Dynamic Processes in a Glassforming Liquid from very Low
to Deep Undercoolings
Vladimir Mihailovich Fokin
Marcio Luis Ferreira Nascimento
Edgar Dutra Zanotto
[email protected]
Vavilov State Optical Institute, St. Petersburg, Russia
Federal University of São Carlos, Brazil
Federal University of Bahia, Brazil
Federal
University
of
• Bahia
MOTIVATION
OUTLINE
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
Federal
University
of Bahia
MOTIVATION1
• 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:
log10 D
Td = 1.1-1.2Tg?
D
D
1/T
Tg
 Which moving units control crystallization? Single atoms or
is it a cooperative movement of “molecules”?
Federal
University
of Bahia
MOTIVATION2
Crystallization theories typically contain a transport
and a thermodynamic term:
I(T) = (K/3)  DI  exp(W*/kBT)
U(T) = (K´/)  DU  [1exp(G/kBT]
transport
thermodynamic
Most authors use viscosity data and the SE / E
k BT
equation to estimate D
DU  DI  D 

Federal
University
of Bahia
(*) from Zanotto‘s thesis
OBJECTIVE
• Our objective is to shed light into the
previous questions by comparing 6 types
of diffusion coefficients in Li2O2SiO2
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.
Federal
University
of Bahia
GLASS DEFINITION1
M. L. F. Nascimento. J. Mat. Educ. 37 (2015) 137-154
Silicon
Oxygen
Crystalline Silica (Quartz, Sand)
Silica Glass
Federal
University
of Bahia
Glass: non-crystalline solid that presents
glass transition phenomenon
M.
L.
F.
Nascimento. J.
Mat. Educ. 37
(2015) 137-154
Volume

GLASS DEFINITION2
Glass transition
temperature
definition Tg:
volume variation
with temperature.
T
g
T
melt
Temp.
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: Li2O2SiO2
ii) only nucleate at surface:
 Silica: SiO2
Federal
University
of Bahia
1. Data Digging &
Analysis of 6 Kinetic
Processes

Crystal growth, viscous flow, conductivity, nucleation time-lag and
self-diffusion coefficients, plus effective diffusion coefficient (the
last is theoretical...)
Federal
University
of Bahia
VISCOSITY DATA
ANALYSIS
Federal
University
of Bahia
VISCOSITY
B
log10   A 
T
Svante Arrhenius
B
log10   A 
T  T0
Gordon Fulcher Gustav Tammann
DIFFUSION
k BT
D 

George Stokes
Albert Einstein
Henry Eyring
Federal
University
of Bahia
log10   2.6234  3388.8 / T  491.05
VISCOSITY
400
600
800
o
T ( C)
1000
1200
1400
14
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
12
Rikuo Ota
log10 (Pa·s)
10
8
6
4
*
Heslin & Shelby
Izumitami
Joseph
*
Matusita & Tashiro
2
0
600
Li2O2SiO2
800
1000
12
Carlos
Gonzalez-Oliver
10
8
6
4
John Bockris
2
0
1200
T (K)
1400
1600
1800
James Shelby
COMPARISON
VISCOSITY MODELS
Federal
University
of Bahia
Experimental data
VFTH: log10   A  B/(T  T0)
14
2
A  2,66234; B  3432,53748; T0  490,70698) :  = 0,02515
Mauro: log10   A  (12  A)[Tg/T]exp[(m/(12  A))  1][(Tg/T)  1]
12
2
A  1,21646; m  43,58019; Tg = 724,44471:   0,02384
log10 (Pa·s)
10
8
VFTH,
Dienes-MacedoLitovitz
and
Mauro’s
proposals are very similar
6
4
2
0
800
1000
1200
T (K)
1400
1600
1800
Federal
University
of Bahia
CRYSTAL GROWTH
DATA ANALYSIS
(The best fittings are shown by a red line)
Federal
University
of Bahia
CLASSICAL CRYSTAL
GROWTH MODELS1
i) Normal (N)
ii) Screw Dislocations (SD)
iii) Surface Nucleation (2D)
Federal
University
of Bahia
i) Normal (N)
CLASSICAL CRYSTAL
GROWTH MODELS2
D
U T   U


 G 
1

exp



 RT 

ii) Screw Dislocations (SD)
G Tm  T
f 

4Vm 2Tm
iii) Surface Nucleation (2D)
k BT
DU  D 

U T   f
DU


 G 
1

exp



RT



C  C N S , , G, T 
U T   C
DU
Z 

exp



2

 TG 
Only one adjustable parameter for all: l
2D: unknown surface energy 
 Vm  2
Z
3k B
Federal
University
of Bahia
GROWTH
o
T ( C)
-4
400
DU
U T   f

500
600
700
SD growth
800
900
 = 0.35Å
1000
10
-4
10
Tg = 454oC
-5
10
-5
10
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.
-6
10
1.1Tg
-7
10
-8
U (m/s)

 G 
1  exp  RT 



10
-9
10
-10
10
-11
10
-12
10
-13
Peter James
-6
10
-7
10
-8
10
-9
10
-10
10
Michael Weinberg
-11
10
-12
10
-13
10
10
700
Li2O2SiO2
800
900
1000
T (K)
1100
1200
1300
Joachim Deubener
Federal
University
of Bahia
TIME LAG DATA
Federal
University
of Bahia
NUCLEATION & TIME LAG1
Li2O2SiO2
• “Model” glass that shows nucleation in
volume.
  3tind
Fixed Temperature
NV
tangent = NV / t = I
Fokin, Zanotto,
Yuritsyn & Schmelzer
352 (2006) 2681
tind
time
m



NV t  t
2
 1
t 


  2 2 exp  m 2
I  tind tind 6
tind 
m 1 m

Collins-Kashchiev
20m
40min at 455oC + 14min at 620oC
I = nucleation rate [m3s1]
Federal
University
of Bahia

NUCLEATION & TIME LAG2
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...
*

DI
W 

I  N 0 2 exp 

 k BT 
3
16

*
W 
2
3GV
• 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.
Josiah W. Gibbs Max Volmer
Iwan Stranski
Rostislav
Kaischew
Yakov Zeldovich
Richard
Becker
David
Turnbull
Federal
University
of Bahia
Li2O2SiO2
TIME LAG
760
750
T (K)
740
730
720
710
700
14
690
14
Peter James
ln  (s)
ln   79.185  63602.287 / T
12
12
10
10
Fokin et al.
James
Zanotto
Tuzzeo
8
6
Vladimir Fokin
1.30x10
8
6
-3
1.35x10
-3
1.40x10
1
1/T (K )
-3
1.45x10
-3
D 
 W* 
 G 
DI
 U T   f U 1  exp 

I  N 0 2 exp 
 

 RT 
 k BT 
NUCLEATION
&
G
ROWTH
T (K)
700
10
800
900
1000
1100
1200
1300
9
3 1
I (m s )
1.1Tg
10
10
SD growth
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
-9
10
-10
10
-11
10
-12
10
-13
8
Li2O2SiO2
7
Tg
700
800
900
1000
T (K)
1100
1200
Tm
1300
 = 0.35Å
Fokin
James
Tuzzeo
Zanotto
Zeng
U (m/s)
Federal
University
of Bahia
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
Federal
University
of Bahia
CONDUCTIVITY DATA
ANALYSIS
Federal
University
of Bahia
CONDUCTIVITY
Ana Candida Rodrigues
20001500
1
1000
T (K)
Jean Louis Souquet
500
1
Tg
0
Li2O2SiO2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
log10  ( cm )
-2
1
-1
1
-1
0
-9
5.0x10
-9
-4
1.0x10
-3
1.5x10
-3
2.0x10
-3
1
1/T (K )
2.5x10
-3
3.0x10
-3
3.5x10
-3
John Bockris
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
S D
C
IFFERENT
Federal IX
University
of BahiaOEFFICIENTS
DIFFUSION
k BT
D 

D from viscous flow:
Eyring Equation
U
(for normal or screw
DU 
f 1  exp G / RT  dislocation growth)
D from
growth:
D from
conductivity:
k BT
D 
N Li e 2
D from time- lag:
Nernst-Einstein
Equation
80 k BTVm2
D 
3 G 22 
Dcation : measured self-diffusion coefficients
D ‘effective’:
Deffective
1
1


xi
xLi xO xSi
D D D D
i
Li
O
Si
Federal
University
of Bahia

2. Results for
Li2O2SiO2
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 Li2O2SiO2
T (K)
2000
-8
1500
1000
500
Tg
D
-10
ONLY DU
DU
-14
2
log10 D (m /s)
-12
-16
-18
-20
-22
Tm
6.0x10
-4
8.0x10
-4
1.0x10
-3
1.2x10
-3
1.4x10
1
1/T (K )
-3
1.6x10
-3
1.8x10
-3
2.0x10
-3
Federal
University
of Bahia
Li2O2SiO2
DIFFUSIVITY2
DU  D for T > 1.1Tg
D < DU low T
T (K)
2000
-8
1500
1000
500
Tg
-10
DU & D
DU
D=kBT/
-14
2
log10 D (m /s)
-12
-16
-18
Breakdown
at 1.1Tg ?
-20
-22
Tm
6.0x10
-4
8.0x10
-4
1.0x10
-3
1.2x10
-3
1.4x10
1
1/T (K )
-3
1.6x10
-3
1.8x10
-3
2.0x10
-3
Federal
University
of Bahia
DIFFUSIVITY3
T (K)
2000
-8
1500
Li2O2SiO2
DU  D  D* for T > 1.1Tg
DLi > DO > DU > D low T
1000
500
Tg
-10
D
*
DU, D & D’s
DLi
DU
D=kBT/
-14
DLi: Beier & Frischat
2
log10 D (m /s)
-12
DO: Takizawa et al.
-16
DO: Sakai et al.
-18
*
D : Kawakami et al.
DO: Takizawa et al.
-20
-22
DO: Sakai et al.
Tm
6.0x10
-4
8.0x10
-4
1.0x10
-3
1.2x10
-3
1.4x10
1
1/T (K )
-3
1.6x10
-3
1.8x10
-3
2.0x10
-3
Federal
University
of Bahia
Li2O2SiO2
DIFFUSIVITY4
n = 51028 m3
T (K)
2000
-8
-10
1500
1000
500
Tg
D
D
*
DU, D, D’s & D
DLi
DU
-12
D=kBT/
-14
D
2
log10 D (m /s)
k BT
D  2 
ne
DLi: Beier & Frischat
DO: Takizawa et al.
-16
DO: Sakai et al.
-18
*
D : Kawakami et al.
D
DO: Takizawa et al.
-20
-22
DO: Sakai et al.
Tm
6.0x10
-4
8.0x10
-4
1.0x10
-3
1.2x10
-3
1.4x10
1
1/T (K )
-3
1.6x10
-3
1.8x10
-3
2.0x10
-3
Federal
University
of Bahia
Li2O2SiO2
DIFFUSIVITY5
80 k BT
D 
3 GV2 2 
T (K)
2000
-8
-10
1500
1000
500
Tg
D
D
*
DU, D, D’s , D & D
DLi
DU
D=kBT/
-14
D
2
log10 D (m /s)
-12
DLi: Beier & Frischat
DO: Takizawa et al.
-16
DO: Sakai et al.
-18
*
D : Kawakami et al.
D
DO: Takizawa et al.
-20
-22
Tm
6.0x10
-4
8.0x10
D
DO: Sakai et al.
-4
1.0x10
-3
1.2x10
-3
1.4x10
1
1/T (K )
-3
1.6x10
-3
1.8x10
-3
2.0x10
-3
Federal
University
of Bahia
DIFFUSIVITY6
-10
1500
1000
500
 = 0.1584 J/m2
Tg
D
D
 W* 
I2

DI 
exp
N0
 k BT 
N0 = 9.781027 m3
T (K)
2000
-8
Li2O2SiO2
*
DU, D, D’s , D , D & DI
DLi
DU
D=kBT/
-14
D
2
log10 D (m /s)
-12
DLi: Beier & Frischat
DO: Takizawa et al.
-16
DO: Sakai et al.
-18
*
D : Kawakami et al.
D
DO: Takizawa et al.
-20
-22
Tm
6.0x10
-4
8.0x10
D
DO: Sakai et al.
-4
1.0x10
-3
1.2x10
-3
1.4x10
1
1/T (K )
-3
1.6x10
-3
1.8x10
-3
2.0x10
DI
-3
Federal
University
of Bahia
DIFFUSIVITY7
log10 DU = 5.5508 + exp(352.4 kJ/RT)
1
1/T (K )
1.2x10
-17
-3
Li2O2SiO2
1.3x10
-3
1.4x10
-3
log10 D = 18.8486 + exp(559.5 kJ/RT)
1.5x10
-3
1.6x10
-3
-3
1.7x10
-17
o
Tg = 454 C
DU  D  DI >> D
-18
-19
-19
-20
-20
2
log10 D (m /s)
-18
-21
-21
DO
-22
-22
1.1Tg
-23
1.2x10
-23
-3
1.3x10
-3
1.4x10
-3
1.5x10
1
1/T (K )
-3
1.6x10
-3
1.7x10
-3
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.
DU
D
D
DI
DO: Takizawa et al.
DO: Sakai et al.
Federal
University
of Bahia
COMPARISON WITH
SIMULATION: MD
D=kBT/ : 0.3Å
DLi: MD
José Pedro Rino
DSi: MD
-8
DO: MD
Deff
2
log10 D (m /s)
-9
D=kBT/ : 2.7Å
-10
L. G. V. Gonçalves, J. P.
Rino. J. Non-Cryst.
Solids 402 (2014) 91-95
-11
-12
-13
4,0x10
-4
5,0x10
-4
6,0x10
-4
7,0x10
-4
1
1/T (K )
8,0x10
-4
9,0x10
-4
1,0x10
-3
Federal
University
of Bahia

3. Results for SiO2
Crystal growth, viscous flow, silicon and oxygen self-diffusion in a
silicate glass that does not display nucleation in volume at
laboratory time-scales
M. L. F. Nascimento, E. D. Zanotto. Diffusion Processes in Vitreous Silica Revisited. Phys.
Chem. Glasses 48 (2007) 201-216
Federal
University
of Bahia
DIFFUSIVITY1 SiO2
T (K)
2200
2000
1800
1600
1400
1200
-15
Tg = 1451 K
-16
DU: Wagstaff
-17
2
log10 D (m /s)
-18
Normal growth
-19
 = 2Å
-20
-21
-22
-23
-24
Tm = 2007 K
5.0x10
-4
6.0x10
-4
7.0x10
-4
1
1/T (K )
8.0x10
-4
9.0x10
-4
Federal
University
of Bahia
DIFFUSIVITY2 SiO2
DU  D at T > 1.1Tg
T (K)
2200
2000
1800
1600
1400
1200
-15
Tg = 1451 K
-16
DU: Wagstaff
D: Brebec et al.
-17
2
log10 D (m /s)
-18
Normal growth
-19
 = 2Å
-20
-21
-22
-23
-24
Tm = 2007 K
5.0x10
-4
6.0x10
-4
7.0x10
-4
1
1/T (K )
8.0x10
-4
9.0x10
-4
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DIFFUSIVITY3 SiO2
T (K)
2200
2000
1800
1600
1400
1200
-15
Tg = 1451 K
-16
DU  D at T > 1.1Tg
DSi  Dh at T < 1.1Tg
No breakdown
with U till 1.1Tg
and with DSi T < Tg
DU: Wagstaff
D: Brebec et al.
-17
DSi: Brebec et al.
2
log10 D (m /s)
-18
Normal growth
-19
 = 2Å
-20
-21
-22
-23
-24
Tm = 2007 K
5.0x10
-4
6.0x10
-4
7.0x10
-4
1
1/T (K )
8.0x10
-4
9.0x10
-4
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DIFFUSIVITY4 SiO2
T (K)
2200
2000
1800
1600
1400
1200
-15
Tg = 1451 K
-16
D: Brebec et al.
DSi: Brebec et al.
DO: Kalen et al.
-18
2
No breakdown
with U till 1.1Tg
and with DSi T < Tg
DU: Wagstaff
-17
log10 D (m /s)
DU  D at T > 1.1Tg
DSi  Dh at T < 1.1Tg
Normal growth
-19
 = 2Å
-20
-21
-22
-23
-24
Tm = 2007 K
5.0x10
-4
6.0x10
-4
7.0x10
-4
1
1/T (K )
8.0x10
-4
-4
M. L. F. Nascimento, E. D.
Zanotto Phys. Rev. B 73 (2006)
9.0x10
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
4. Conclusions
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INITIAL QUESTIONS
i) Does the SE/ E equation breakdown at some low
enough T?
log10 D
Td ~ 1.1-1.2Tg?
D
D
Tg
1/T
ii) Which moving units control crystallization? Single
atoms or is it a cooperative movement of molecules
(Deff)?
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log10 D
SKETCH
Td ~ 1.2-1.1Tg
DO?
DSi
DLi
D
DO
1/T
Tg
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log10 D
CONCLUSION: SKETCH
DU (surface) > Deff (volume)
Td ~ 1.2-1.1Tg?
DO?
DSi
DLi
Deffective  D
( = 2.7Å)
D
Time lag
DO
1/T
Tg
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CONCLUSIONS1
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 T1.1Tg 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.
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CONCLUSIONS2
 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 ~1028m3) 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.
Acknowledgments
THANK YOU!!!!
FROM LAMAV BASIS AT SALVADOR, BAHIA
Boipeba
Salvador
Maraú
Federal
University
of Bahia
Federal
University
of Bahia

Dedução das Leis
de Fick
A. E. Fick, Federal
Phil. Mag. 10 (1855) 30
A. E. Fick, Ann. Phys. 170 (1855) 59
University
of Bahia
Adolf Eugen Fick (1829-1901), médico e fisiologista alemão
Federal
University
of Bahia
Difusão em Sólidos1
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 :
área A
n
n
vt
vt
x
área A
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n
vt
n
vt
Difusão em
Sólidos2
é possível contar o número de
x partículas que atravessam A :
 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.
 Considere ainda o número que atravessa uma superfície A em um
tempo t dado pelo número de partículas que estão a uma distância
vt de A.
(onde v é a velocidade molecular real )
nvt
número de partículas que atravessam a superfície da esquerda para direita:
n número de partículas por unidade de volume à esquerda do plano A.
n número de partículas por unidade de volume à direita do plano A.
nvt
número de partículas que atravessam da direita para a esquerda:
área A
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n
vt
Difusão em
Sólidos3
n
vt
x
 Definindo J como uma corrente molecular,
ou a densidade de corrente que atravessa o J  n vt  n vt  n  n v


plano A, então J corresponde ao fluxo total
2t
2
de partículas por unidade de área por
unidade de tempo:


 Considere ainda uma distribuição espacial das n partículas por uma
função contínua de x, y e z .
dn
dn
 Assim pode-se entender n(x, y , z) como a
densidade de partículas em um pequeno n  n  dx x  dx 2l
elemento de volume xyz centrado em (x
, y , z). Em termos de n podemos expressar (onde l é o livre caminho médio)
a diferença:
dn
v
 Substituindo:J  
 2l 
dx
2
ou
dn
J x  lv
dx
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n
vt
Difusão em
Sólidos4
n
vt
x
 Em três dimensões o resultado acima diferencia
de um fator 1/3, devido à isotropia do espaço.
Logo uma resposta melhor é:
ou
lv dn
Jx  
3 dx
dn
J x  D
dx
 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:
1
2 dn
J x   mv 
3
dx
mas pelo Princípio
da Equipartição da
Energia tem-se que:
1
3
mv 2  k BT
2
2
dn
J x  k BT
dx
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
PRIMEIRA LEI DE FICK
área A
Difusão em
Sólidos5
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D  k BT
 E se ainda for considerado um fluxo
n de partículas com carga elétrica, a  
Equação da Continuidade diz que, em   J   
t
relação ao princípio de conservação
da carga:
    
  J   , ,   J x ,0,0
 x y z 
n

d 
dn 
  Jx    D 
t
x
dx 
dx 
 No caso em que D não depende da concentração n, obtém-se a SEGUNDA LEI
DE FICK:
dn
d 2n
D 2
dt
dx
SEGUNDA LEI DE FICK
Sendo:
Adolf Eugen Fick
(1829-1901), médico
e fisiologista alemão
Federal
University
of Bahia

Resumo das Leis
de Fick
Federal
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of Bahia
Primeira Lei de Fick1
 O fluxo da impureza na direção x é proporcional ao
gradiente de concentração n nesta direção.
A
n
dn
dx
dn
J x  D
dx
Jx : Fluxo de átomos
através da área A
[átomos/m2s]
D : coeficiente de difusão
x ou difusividade [m2/s]
Federal
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
Primeira Lei de Fick2
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.
na
nb  na
dn
J x  D
 D
dx
xb  xa
J
nb
na
nb
xa
xb
posição x
Federal
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Segunda Lei
de Fick
 A taxa de variação da
concentração com o tempo, é
igual ao gradiente do fluxo:
dn d  dn 
 D 
dt dx  dx 
Exemplo de difusão: oscilações
próximas de posições de equilíbrio
permitem saltos eventuais e
aleatórios para as vacâncias vizinhas
 Se a difusividade não depende de x:
dn
d 2n
D 2
dt
dx
 Esta equação diferencial de segunda ordem só pode
ser resolvida se forem fornecidas as condições de
fronteira.