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Università Ca’ Foscari Venezia
Diffusione nei solidi: teoria e applicazioni a materiali avanzati
Prof. Elti CattaruzzaDepartment of Molecular Sciences and Nanosystems, Università Ca’ Foscari Venezia,via Torino 155/b, 30172 Venezia-Mestre, Italy
Why to study diffusion?
– The movement of atoms is favoured by thermal energy
– Such a movement can be described by equations
– Suitable modifications of materials can be proposed and realized
– Material properties can be improved
An enlightening example
Copper-nickel pair before a thermal treatment at high temperature
Copper (nickel) atoms are present only in the left (right) region
Atomic concentration as a function of the position inside the sample
←
←
←
...after a thermal treatment below melting temperature
In the central region a Cu-Ni alloy has formed
Copper and nickel atoms migrated in the opposite region
Atomic concentration as a function of the position inside the sample
←
←
←
What is “diffusion”?
DIFFUSIONDIFFUSION :: the movement of atoms or molecules in a material
To understand the diffusion in solids, we need to know something about:
• imperfections in solids (in crystals)
• thermal atomic vibrations
Imperfections in crystals
– Point defects
– Line defects (dislocations)
– Surfaces
– Grain boundaries
The most important imperfections for the diffusion are: point defectspoint defects
Point defects
a) a missing atom within a crystalb) two (three, four, ...) missing atomsc) a missing ion-pair of opposite charge
d) an extra atom within a crystal structuree) a displaced ion from the lattice
Thermal atomic vibrations
– For T>0 K, atoms have kinetic energy (i.e. vibrations)
– Atoms have different energies (i.e. distribution)
– The fraction n/Ntot of atoms with energy greater than E (for E»E) is:
_
eNkTE
tot
n −∝
k is the Boltzmann’s constant (k=1.38×10-23J/K)Units of E: Joules
E
kinetic energy
T1
T2
T3
T1< T2 < T3Prob.
n/Ntot
... an important consequence of the thermal energy
– There is a (small) fraction of atoms having very large energies:if this energy is large enough, some atoms can break their bondsand jump to new locations, thus originating vacancies...
The minimum energy required is called activation energy Q (or E)
““DIFFUSIONDIFFUSION””
(units of Q or E can be J/atom, eV/atom, J/mole, or calories/mole)
MeNkTE
tot
n −∝
Diffusion mechanisms
An atom leaves its lattice site to fill a nearby vacancy (thus creating a new vacancy)←
When a small interstitial atom is present, the atoms moves from one interstitial site to anoth←
A substitutional atom leaves its normal lattice site and enters an interstitial position←
Atoms move by a simple exchange mechanism or by a ring mechanism←
a), b) are the main ones
Activation energy for diffusion
low activation energy Q
diffusion favoured
⇓
…a more realistic sketch of the first example shown (Cu-Ni pair)
Point defects are very important for diffusion, allowing the atoms movement
Temperature is very important for diffusion, increasing the movement of atoms and the creation of new vacancies
The diffusion equations
Diffusion depends on time. How can we define the atoms flux during diffusion?
flux, J: number of atoms passing per unit time through a plane of unit area (⊥ to the diffusion direction)
Units of J: atoms/m2 sec
Fick’s first law (rate of diffusion)
The flux J of atoms is proportional to the concentration gradient
J = atoms flux (atoms/m2•sec)D = diffusivity (m2/sec)C = concentration (atoms/m3)
Stationary diffusion: J(x,t)= J(x)
dxdCDJ −=
Fick’s first law
…factors influencing the diffusion coefficient (diffusivity) D
The diffusivity D depends on:1) the nature of the solute atoms
2) the nature of the solid structure
3) the temperature
Small atoms have a higher diffusivity than large onesAtoms can be neutral or charged
Atoms diffuse more readily in weakly bonded solidsAtoms diffuse more readily in low packed solids
Higher temperatures provide higher diffusivities
D0 is the proportional constant k=1.38×10-23J/KE is the activation energy per atom (Joules)eDD kT
E
TE −=),( 0
(an Arrhenius-type equation)
…temperature and the diffusion coefficient D
eDD kTE
TE −=),( 0 eDD RTQ
TQ −=),( 0
D0 is the proportional constant R is the gas constant (R=8.314 J/mole•K, or 1.986 cal/mole•K)Q is the activation energy (J/mole, or cal/mole)
The logarithm of the diffusivity D:
kTEDD −= 0lnln
RTQDD −= 0lnln
Data of D are usually plotted in Arrhenius-type plots
Arrhenius-type plot of diffusivity D
kTEDD −= 0lnln
RTQDD −= 0lnln
Fick’s 2nd law (composition profile)
In most cases, the diffusion is a dynamic process!
C(x) =C(x,t) (the concentration C of diffusing atoms changes with time, so does the flux J)
From the Fick’s first law:
⎟⎠⎞
⎜⎝⎛−=
xCD
xxJ
∂∂
∂∂
∂∂
The equation of continuity:
tC
xJ
∂∂
∂∂
−=
…still Fick’s second law
Being , the final differential equation is:
⎟⎠⎞
⎜⎝⎛=
xCD
xtC
∂∂
∂∂
∂∂
Fick’s second law2
2
xCD
tC
∂∂
∂∂
=
...and if the diffusivity D does not depend on composition:
tC
xJ
∂∂
∂∂
−=
(...for the ideal dilute case Nernst-Einstein have shown that D=ũkT)
ũ = mobility of diffusing species = average velocity/unit force
…solving the Fick’s second law
The solution of the Fick’s 2nd law depends on the boundary conditions!
A practical case:
the concentration value of the diffusing atoms at the surface of the material is a constant (Cs).
Solution:Solution:
⎟⎠
⎞⎜⎝
⎛−−=DtxerfCCCtC ssx 2
)()( 0
(Cx(t) is the concentration at depth x, at time t)
the initial concentration of the diffusing atoms in the material is uniform (C0);
t > 0, C=Cs for x=0 and C=C0 for x=∞
t = 0, C=C0 for any positive x
Cx(t)
x
Error function and composition profile
Dtxzdyyzerf
ze 2
,2)(0
2=−= ∫π
In non steady-state diffusion, the concentration depth profile of diffusing species changes with time
The thickness of the region interested by the diffusion increases with (Dt)1/2
…a practical case: the carburizing of steels
A thermal treatment of a solid in controlled atmosphere exhibits the describedboundary conditions!
Carburization of steel:Carburization of steel: thermal treatment of steel at high temperature in a carbon-rich atmosphere (often with CH4 )
(the induced formation of iron carbide on the surface increases the hardness of the material)
The ionThe ion--exchange exchange (an example of diffusion in silicate glasses)(an example of diffusion in silicate glasses)
Silicate (e.g. SiO2-based) glass structure
Crystalline silica: long-range structural orderVitreous silica: limited structural order (max. 3 nm)Silicate glass: SiO2 with other oxides
crystal glassglass with modifiers
…the ion exchange in silicate glassesEbond ≈ 1 eV (for alkaline ions) ...easy diffusion!
Process parameters:Process parameters:
1. bath composition2. bath temperature3. exchange time
An alternative method:Field-assisted solid-stateion exchange (FASSIE)
* A. Quaranta, E. Cattaruzza, F. Gonella, “Modelling the ion exchange process in glass: Phenomenological approaches and perspectives”, Mat. Sci. Eng. B 149 (2008) 133–139
… (not conventional) ion exchange in silicate glasses
metal contact
metal contact
glass substrate
isolation
+ –
µAit
oven
deposited film
RF magnetron sputtering
Metallic film 50-200 nm thick
Au metallic film 200 nm thick
(ohmic contact)
PARAMETER RANGE
(temperature) T 100 to 500 °C
(electric field) E 10 to 500 V/mm
(process time) t minutes; hours
Apparatus configurationSilicate glass slide
−−
Met+
Na+
O−
+
E
+
…the ion exchange in silicate glasses
Ion diffusion in glassIon diffusion in glass• transport of charge entirely due to metallic ions• two ionic diffusing species (A and B) with the same charge• vacancy (V) concentration nearly constant (equilibrium)• (local) thermodynamic equilibrium
• diffusion by site changes A-B, A-V, B-V
Hypothesis
0=++ VBA JJJ
Diffusion towards “x” direction
xTL
xTLJ
xTL
xTLJ
BBBABAB
BABAAAA
∂∂µ
∂∂µ
∂∂µ
∂∂µ
−−=
−−=
L: phenomenological parametersµ: chemical potentialT: (absolute) temperature
…the ion exchange in silicate glasses
Nernst-Planck equation(s)
),(),(),()(ln)(ln1),(
txctxEutxx
cF
DtxJ AAA
A
AAA +⎟⎟
⎠
⎞⎜⎜⎝
⎛+−=
∂∂
∂γ∂
D: diffusion coefficientγ: activity coefficientF: ion atomic fractionu: ion mobilityc: concentration
E: electric (local) field⎟⎟⎠
⎞⎜⎜⎝
⎛+=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
A
AB
A
AAA
B
AB
A
AAA
cL
cL
kTqu
cL
cLkD
q: ion charge
kTqDu A
A =? No NernstNo Nernst--Einstein:Einstein:
kTfqDu A
A =
f: correlation coefficient
…the ion exchange in silicate glasses
Alternative (equivalent) description
A–B (or A–V): two equilibrium level system with an energy barrier, ∆
eDD kTT∆
−=∆ ),( 0
Kinetic reaction at the surface…A*, B*= liquid phase (salt); A, B =solid phase (glass) ** BABA +=+
),(),(),()(ln)(ln1),(
),(),(),()(ln)(ln1),(
txctxEutxxc
FDtxJ
txctxEutxxc
FDtxJ
BBB
B
BBB
AAA
A
AAA
+⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
+⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
∂∂
∂γ∂
∂∂
∂γ∂…then diffusion
…the ion exchange in silicate glasses
Charge balanced → E=E(FA, γA, D)
xc
FFDFDDDtxJ A
A
A
BBAA
BAA ∂
∂∂
γ∂⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−=
)(ln)(ln1),(
qA=qB=q
Final equation to solve:
Hypotheses usually done: • cA = constant at the surface =cS (goodgood)• cA= 0 in the glass at the beginning (goodgood)• D does not depend on x (STRONGSTRONG)˜
Solution: ⎟⎠
⎞⎜⎝
⎛=⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−=Dtxerfcc
Dtxerfctxc ssA 22
1),(
xcDtxJ A
A
~),(∂∂
−=
⎟⎠⎞
⎜⎝⎛=
xcD
xtc AA
~∂∂
∂∂
∂∂
…the ion exchange in silicate glasses
Ag+– Na+ ion exchange in SLG
By solving the diffusion equation:
Good control of the doping profile
Ag+– Na+ ion exchange in SLG
Refractive index and Ag depth profile
“graded-index”waveguide
…the ion exchange in silicate glasses
Soda-lime glass (SLG)
wt% at. % (approx.)
69.6 SiO2 24.1 Si
15.2 Na2O 10.2 Na
1.1 K2O 0.5 K
6.5 CaO 2.4 Ca
5.1 MgO 2.6 Mg
1.8 Al2O3 0.6 Al
(0.7 others) 59.6 O
Borosilicate glass (BSG)
wt% at. % (approx.)
69.6 SiO2 23.7 Si
8.4 Na2O 5.5 Na
8.4 K2O 3.7 K
9.9 B2O3 5.8 B
2.5 BaO 0.3 Ba
(1.2 others) 61.0 O
Ag+– Na+
Cu+– Na+
SLG BSG
SLG SLG
FASSIE
…the FASSIE in silicate glasses
10-1
100
101
102
103
104
105
0 200 400 600
Ion
yiel
d (c
ount
s/s)
Depth (nm)
OCaSi
AuK
Na
T=400 °C, E=500 V/mm (t=120 min)
SLG matrix
Co2+ Au3+
Er3+
10-1
100
101
102
103
104
105
0 100 200 300 400 500 600 700
Ion
yiel
d (c
ount
s/s)
Depth (nm)
OCaSi
ErK
Na
T=500 °C, E=200 V/mm
…MNCGs (metal nanocluster composite glasses) synthesis
MNs embedded in glass enhance its optical third-order susceptibility (Kerr effect)...
intensity-dependent refractive index(useful for photonics applications)
↵
…two-step MNCGs synthesis by diffusion processes (1st example)
a silicate glass is immersed in a molten salt bath containing the doping atoms; they replace alkali ions of the glass by diffusion
Ion exchange:
the activation energy for the diffusion of ions is usually high!
1. Metal-alkali ion exchange2. Thermal treatment in controlled atmosphere (H2, O2, Ar, …)
…the first step (ion exchange) →waveguide
Silicate glass refractive index: nglassMetal-doped silicate glass refractive index : ndoped-glass
ndoped-glass > nglass
θi θr
θt
n1
n2(n2 > n1)
Snell’s law:n1 sinθi = n2 sinθt
Waveguide! ndoped-glass
nair
nglass (ndoped-glass > nair , nglass)
glass containing Na (and K, Ca, Mg, ...) ions:soda-lime glass
Na+–Cu+ ion exchange in CuSO4:Na2SO4bath at T= 545 ˚C for 10 minutes
thermal treatment in Ar–H2(4%) at T= 160 ˚C for 5 hours
←
←
←
1st step
2nd step
…two-step MNCGs synthesis by diffusion processes (1st example)
depletion of Na,accumulation of Cu
copper diffuses inside the glass both as Cu+ and Cu++ ions
Hydrogen diffusion inside the doped glass induces precipitation and aggregation of copper atoms
1st step
2nd step
…two-step MNCGs synthesis by diffusion processes (1st example)
1. Metal ion implantation2. Thermal treatment in controlled atmosphere (H2, O2, Ar, …)
a silicate glass is bombarded with metal ions accelerated by a potential difference of several kV; they enter the glass and dope it (atoms can also diffuse during the implantation by RED)
Ion implantation:
…two-step MNCGs synthesis by diffusion processes (2nd example)
…two-step MNCGs synthesis by diffusion processes (2nd example)
glass (SiO2, silicate glass, …)
Au+ ion implantation at E=190 keV, F=3×1016 ions/cm2, j≤2 µA/cm2
thermal treatment in air at T= 900 ˚C for 1 hour
←
←
←
1st step
2nd step
gold implanted atoms are dispersed inside the glass
Inside the doped region, the O2diffusion induces the gold atoms to move and aggregate to form metal particles in the nm range of size
1st step
2nd step
surface
…two-step MNCGs synthesis by diffusion processes (2nd example)
…two-step MNCGs synthesis by diffusion processes (3rd, 4th
examples…)
glass (SiO2, silicate glass*, …)
laser irradiation (λ=527 nm, ε=0.5 J/cm2)OR
ion irradiation (1 MeV Xe+, 3×1016 ions/cm2)
←
←
←
1st step
2nd step
Na+–Ag+ ion exchange*OR
Ag+ ion implantation
ion irradiation (1 MeV Xe+, 3×1016 ions/cm2) laser irradiation (λ=527 nm, ε=0.5 J/cm2)
Na+–Ag+ ion exchange
1st step
2nd step
…two-step MNCGs synthesis by diffusion processes (3rd, 4th
examples…)
2nd step
Ion exchanged silicate glasses for solar cells
covering: down-shifting properties
Conversion efficiency of the solar cells
WAFER THIN FILM
about 8% of power falls in the
near-UV region
downshiftingSilicate glasses doped with transition elements such as Ag and Cu are known to be
luminescent materials
Modification of the incoming light spectrum!
… scientific frame…
ion exchange substrate
A+A+
B+B+Main parameters:
1) bath composition2) atm. composition3) Texc and texc
4) … and also subsequent thermal treatments, if needed…
in form of nanoparticles{
… glasses and baths…
Float-glass?SnSn2+2+ contamination!contamination!
• One-side contamination• Sn2+ in the first few microns• Reducing agent: Sn2+ → Sn4+
Salt bathSalt bathCuSO4:Na2SO4 (46:64),T=550°C, t=20 min
AgNO3:NaNO3 (1:99),T=320°C, t=60 min (annealed)
Too muchCu2+, Cu0
Too muchAg0
… change of(Cu) BATH CuCl:ZnClCuCl:ZnCl22 (11:89)
(Ag) IMMERSION Floated (F)Floated (F), not Dipped (D)
{
… optical absorption…
AgNO3:NaNO3 (1:99)T=320°C, t=60 min
(annealed)
CuCl:ZnCl2 (11:89)(as-exchanged)
Ag precipitation favoured by Sn presence
… mainly Cu+SILVERSILVER
COPPERCOPPER
… (Ag) photoluminescence…
Band centered at 450-500 nm: interaction between Ag+ -Ag+ pairs
Band centered around 600 nm: presence of charged few-atoms aggregates
• the downshifting takes place…• …controlling the silver aggregation
… (Cu) PL and XANES…
Band centered around 500 nm: high value of the Cu+/Cu2+ ratio
PL and PLE XANES
… P-against-V cell tests…
GaAs celloutput power
6.2
6.3
6.4
6.5
6.6
6.7
660 680 700 720 740 760
Cu-doped glass (350°C-3h)
Cu-doped glass(350°C-20min)
Ag-doped glass(floating)
Reference pure glass
Pow
er (m
W)
Voltage (mV)
0
2
4
6
8
10
0 200 400 600 800 1000
Cur
rent
(mA
)
Voltage (mV)
Fill Factor = 0.77
ISC
VOC
VMP
IMP
0
1
2
3
4
5
6
7
0 200 400 600 800 1000
Pow
er (m
W)
Voltage (mV)
VMP
…an encouraging result*!
* E. Cattaruzza et al., Sol. Energy Mater. Sol. Cells 130 (2014) 272–280
… silver again…
How to maximize the formation of (charged) few-atoms aggregates
avoiding Ag nanoparticles?Formation Mechanism of Silver Nanoparticles Stabilized in Glassy Matrices
Anne Simo et al., J. Am. Chem. Soc.134 (2012) 18824−18833
…… a threshold Ta threshold Tannann exists, for any given glass!exists, for any given glass!
… new experiments…
Tin-free soda-lime glass composition
Element Atomic %ION EXCHANGEION EXCHANGE
(T=320°C, t=60 min)
AgNO3:NaNO3 (1:99)AgNO3:NaNO3 (0.1:99.9)
ANNEALING (in air)ANNEALING (in air)
380°CT = 410°C
440°C
1 ht = 4 h
16 h
{{
1 %
0.1 %
380°C 410°C 440°C
1h 4h 16h 1h 4h 16h 1h 4h 16has-exc
… 1 mol% Ag-doped glass…
0.4
0.8
1.2
1.6
2Ref. glassas-exc380°C-1h380°C-4h380°C-16h410°C-1h410°C-4h410°C-16h440°C-1h440°C-4h440°C-16h
300 350 400 450 500 550 600 650
Opt
ical
den
sity
(arb
. uni
ts)
Wavelength (nm)
small silver clusters(d < 1 nm)
0
1 107
2 107
3 107
4 107
5 107Ref. glassas-exc380°C-1h380°C-4h380°C-16h410°C-1h410°C-4h410°C-16h440°C-1h440°C-4h440°C-16h
400 450 500 550 600 650 700
PL in
tens
ity (a
rb. u
nits
)
Wavelength (nm)
λexc = 350 nm BEST440°C410°C longer times380°C very long time
WORST
440°C longer times410°C medium time
410°C-16h should be
the best sample
… P-against-V cell test…
0.034
0.035
0.036
0.037
0.038Ref. glassas-exc380°C-1h380°C-4h380°C-16h410°C-1h410°C-4h410°C-16h440°C-1h440°C-4h440°C-16h
0.36 0.38 0.4 0.42 0.44
Pow
er (W
)
Voltage (V)
Si cell
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 0.1 0.2 0.3 0.4 0.5 0.6
Pow
er (W
)
Voltage (V)
• …yield still lower than with the pure glass• best sample: 410°C-16h
3D Contour Plot of Integrale (400-700 nm), emissione, 350 nm, 1% against t (h) andT (°C)
Integrale (400-700 nm), emissione, 350 nm, 1% = Spline
> 9E9 < 9E9 < 8E9 < 7E9 < 6E9 < 5E9 < 4E9 < 3E9 < 2E9
1VA1
1VA2
1VA3 1VA4
2VA 2VA1
2VA2 2VA3
2VA4
0 2 4 6 8 10 12 14 16 18
t (h)
370
380
390
400
410
420
430
440
450T
(°C
)
1VA1
1VA2
1VA3 1VA4
2VA 2VA1
2VA2 2VA3
2VA4
… looking for the best conditions…
3D Contour Plot of Integrale (400-700 nm), emissione, 350 nm, 1% against t (h) andT (°C)
Integrale (400-700 nm), emissione, 350 nm, 1% = Negative Exponential Smoothing
> 9E9 < 9E9 < 8E9 < 7E9 < 6E9 < 5E9 < 4E9 < 3E9 < 2E9
1VA1
1VA2
1VA3 1VA4
2VA 2VA1
2VA2 2VA3
2VA4
0 2 4 6 8 10 12 14 16 18
t (h)
370
380
390
400
410
420
430
440
450
T (°
C)
1VA1
1VA2
1VA3 1VA4
2VA 2VA1
2VA2 2VA3
2VA4
3D Contour Plot of Integrale (400-700 nm), emissione, 350 nm, 1% against t (h) andT (°C)
Integrale (400-700 nm), emissione, 350 nm, 1% = Distance Weighted Least Squares
> 1E10 < 1E10 < 8E9 < 6E9 < 4E9 < 2E9
1VA1
1VA2
1VA3 1VA4
2VA 2VA1
2VA2 2VA3
2VA4
0 2 4 6 8 10 12 14 16 18
t (h)
370
380
390
400
410
420
430
440
450
T (°
C)
1VA1
1VA2
1VA3 1VA4
2VA 2VA1
2VA2 2VA3
2VA4
3D contour plot of 400-700 nm integrated PL intensity (λexc=350 nm, 1% exc. samples)
Most promisingannealing conditions
410°C ≤ T ≤ 420°C
10h ≤ t ≤ 12h
… PL quantum yield…
Absolute fluorescence quantum yield (number of emitted photons / number of absorbed photons)
λexc=350 nm
* E. Cattaruzza et al., Ceramics International 41 (2015) 7221–7226