Transcript Document

MATGEN IV
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Introduction to Interfaces and Diffusion
1. Forward
2. Interfacial Thermodynamics
2.1 Interfacial excess properties
2.2 Gibbs adsorption equation
3. Statistical Thermodynamic Models of Segregation
3.1 Usefulness of statistical models
3.2 The monolayer model
3.3 Examples of trends
3.4 Integration of the Gibbs adsorption isotherm
4. Anisotropy of Interfacial Properties
5. Grain Boundaries
6. Interfacial Equilibrium
6.1 Equilibrium of a GB with a surface
6.2 Equilibrium of second phases at GBs and wetting (prewetting)
7. Diffusion in Solids – An Overview
7.1 Introduction
7.2 Mathematical description of (Fickian) diffusion -- apologies
7.3 Atomic mechanisms of diffusion in alloys
Forward
Interfaces are regions of a microstructure that separate phases which
differ in structure and/or in composition. (Exception: GBs)
1000
1100
1000

L
900
 Fe C

800
T, °C
T, °C
900
3
 L
 L
800


779°C
700

600

700
 Fe C
500
3
600
0
0.2
0.4
0.6
0.8
wt % C
1
1.2
1.4
400
0
20
40
60
80
wt % Cu
liquid-vapor interface (= liquid surface), solid-liquid interface, solidsolid interface (= interphase boundary, interphase interface)
100
Interfacial Thermodynamics
Follow Gibbs. Interfacial energy is denoted by  (with typical units of
[mJ/m2]). It is defined as the reversible work needed to create unit area
of surface, at constant temperature, volume (or pressure), and
chemical potentials.
Use liquid-vapor surface as an example, i.e., consider a system
composed of a liquid and a vapor phase at equilibrium, separated by
an interface.
dividing surface, hypothetical system
Interfacial excess properties
Examples:
Es = E - E' – E''
interfacial excess internal energy, Es:
and interfacial excess no. of moles of component i: nis = ni - ni' - ni"
V = V' + V''
Exception: no interfacial excess volume,
Internal energies of phases ' and '' are written (as usual):
dE TdS  PdV   
i dni
i
dE TdS  PdV   
dni
i
i
and interfacial excess internal energy (no volume) is written:


dEs  TdSs  dA is dnis
 


i
s

dE  T dS  dS  dS  P dV  dV   dA    i dni dni dnis
i

dE  TdS PdV  dA i dni
i

All other thermodynamic properties can be written from the previous
relation: e.g. Gibbs free energy, G = E + PV - TS
dG  dE  PdV  VdP  TdS SdT  SdT  VdP  dA  i dni
i
Note! The surface excess quantities are arbitrary, since they depend
on precisely where the dividing surface is located within the diffuse
region associated with the interface. We shall see later how this issue
may be addressed.

Gibbs adsorption equation and isotherm
The expression for dEs, is written only in terms of extensive independent
variables (dSs , dA, dnis), while the intensive variables (T, , i) are constant
in the equilibrated system. This makes integration of dEs straightforward:
E s  TSs  A  i nis
i
Re-differentiating yields:
dEs  TdSs  S s dT  dA Ad  i dnis  nis di

i
Comparing with the expression for dEs, we conclude:
S s dT  Ad   nis di  0
i
We now define the following specific interfacial excess quantities:
ss  S s / A
and
i  nis / A
where
i is referred to as the adsorption of component i
d  s s dT  i di
and obtain:
Gibbs adsorption equation
i
At constant temperature, this simplifies to the Gibbs adsorption
isotherm. For a two-component system, it can be written:
d  1d1  2d2
 be simplified further by the use of the Gibbs-Duhem equation:
This may

n1d1  n2d2  0
It is conventional to take components 1 and 2 to be the solvent and
solute, respectively, and to eliminate 1 from the adsorption equation:


d  n2
 1  2 
d 2  n1

The l.h.s. of the above equation is measurable, therefore the r.h.s.
cannot be arbitrary.

Approximate forms of the Gibbs adsorption isotherm
The chemical potential may be expressed as: 2= 2° + kT ln a2,
where a, is the activity and ° is the standard state chemical potential.
For ideal solutions, a2 = x2, where x2 is the mole or atom fraction. For
dilute solutions, a2 = kox2, where ko is Henry's Law constant. In both of
these cases, d2 = RT d(ln x2). Thus, in both those cases the Gibbs
isotherm may be written:
 n2

1 d
 1  2 
RT d lnx2  n1

In particular, in dilute solutions, n'2 << n'1, so that we can write:
1
d
 2
RT d lnx2

This is the most commonly used form of the Gibbs adsorption isotherm.

3. Statistical Thermodynamic Models of Segregation
The Gibbsian approach is not always easy to use
•Neither , nor its variation with composition, are easy to
measure in solids.
•2 is difficult to measure directly, because interfacial
composition profiles can extend some distance from the
interface, and one must determine the composition profiles of
both components in the most general case.
•Gibbsian thermodynamics do not provide a relationship
between 2 and 2. Without such information, the Gibbs
adsorption isotherm cannot be integrated.
These factors have led to the development of various models which
can overcome some of these problems. Here we shall briefly describe
one of the simplest of these models.
Monolayer (surface) segregation model (regular solution approximation)
 Eseg 
x

e xp

s
1 x
1 x
 RT 
xs
xs and x are the atom fractions of the solute in the surface monolayer
  is the energy of segregation, i.e.
and the bulk, respectively, and
seg
the energy change resulting from exchanging a solute atom in the
bulk with a solvent atom at the surface.


Eseg  (  B   A )   2 z ( x  x s )  z v ( x  12 )  Eel
A and B are the surface energies of the pure components (B=solute), 
is the area per mole in the monolayer,  = AB – (AA + BB )/2 is the
regular solution constant, ij are bond energies between i-j pairs of
v
atoms, z is the in-plane coordination and z is half of the out-of plane
bonds made by an atom, and Eel is the elastic strain energy of a solute
atom.
Typical results from this type of model
15
1
x = 0.01; E = 2727 J/mol
el
monolayer
multilayer
2
 = 1650 mJ/m ;  = 918 mJ/m
A
10
s
ln(x /1-x )
0.6
s
layer atom fraction
0.8
2
B
0.4
x (bulk) = 0.1
5
0
 = 562 J/mol
 = 0 J/mol
 = -562 J/mol
0.2
-5
0
1
2
3
4
5
6
0
7
0.0005
0.001
0.0015
0.002
0.0025
1/T(K)
layer number
Also, the relationship between xs and  is:
  (x s  x) / 
From this relation, and the one between xs and x, it is possible to
integrate the Gibbs adsorption isotherm and obtain:

0.003
0.0035
  
RT

ln(1 kx )  x
where k is a surface enrichment factor given by
x s  kx/(1 kx)
1500

1400

2
 (mJ/m )
1300
1200
1100
1000
900
0. 0001
T = 1000 K
T = 800 K
T = 600 K
T = 400 K
0. 001
0. 01
0. 1
1
x
Note! The surface energy is not proportional to the adsorption, rather it
is the slope, d/d, that is proportional to adsorption.
Anisotropy of Interfacial Properties
Example: energies of (100) and (111) fcc surfaces calculated by nearest
neighbor bond model
Each atom in (100) surface has 4 broken bonds. But breaking 4 bonds
creates 2 surfaces. Number of atoms per unit area of surface is 2/a2. If
energy of broken bond is  (= - AA) then:
2 (100) = 4  (2/ a2), or (100) = 4 /a2
For (111) surface, each atom has 3 broken bonds. But breaking 3
bonds creates 2 surfaces. Number of atoms per unit area of surface is
4/(3 a2). Then:
(111) = (3/2)  [4/(3 a2)] = 23  /a2 ~ 3.46 /a2
This approach can be used to compute the relative energies of
surfaces of all possible (hkl) orientations in fcc metals (at 0 K)
Qu i c k T i m e ™ a n d a
T I F F (Un c o m p re s s e d ) d e c o m p re s s o r
a re n e e d e d to s e e th i s p i c t u re .
Data on anisotropy of surface energy can be used to determine the
equilibrium crystal shape (ECS) by means of the "Wulff construction"
Conversely, studies of ECS can yield information on anisotropy of s
Examples: ECS of pure Cu and of Bi-saturated Cu at ~ 900°C (with
monolayer of adsorbed Bi at the surface) illustrates effects of
segregation on ECS
Cu
Bi-saturated Cu
Example: scanning electron microscope image of a Bi-saturated Cu
"negative" crystal
Grain Boundaries (GBs)
Special type of interface in single phase materials. Play important role
in properties of poly-crystalline materials.
Microstructure of tetragonal TiO2 displaying EBSD contrast
Schematic of GB with solute segregation
Orientation space of GBs is 5-d (compared with surfaces that have
2-d orientation space). 5-d space often described by 3 Euler angles +
vector perpendicular to GB plane.
Alternative description of 5-d space: interface plane scheme
Energies of GBs for simplified orientation space (a: symmetric tilt GB,
b: symmetric twist GB
e.g. energy of symmetric tilt GB (Read and Shockley):
GB = B[A – ln()]
Variation of GB energy in a more general orientation space (fcc
metals)
900
plateau region
2
(mJ/m )
850
800
700
650
0
10
20
30
40
60
GB
twist angle (°)
50
 plateau (arbitrary units)
Read-Shockley trend
750

GB
8
7
6
5
4
3
3
3.5
4
mean surface energy (arbitrary units)
Note! Anisotropy of GB energy is larger than that of surface energy
4.5
Example: segregation at GBs of different orientations (multilayer model)
0.07
Pt-1at%Au
T = 1000K
(530) - (530)
(530) - (511)
(530) - (221)
(530) - (111)
planar Au atom fraction
0.06
0.05
0.04
0.03
0.02
0.01
0
-1.5
-1
-0.5
0
0.5
1
distance across GB (lattice constants)
1.5
Interfacial Equilibrium
GB with surface
GB = (hkl)1 cos() + (hkl)2 cos()
or for isotropic surface:
GB = 2s cos()
Example: AFM image of GB grooves at pure Cu surface
Second phases at GBs and wetting
For a liquid phase at a GB: GB = SL1 cos() + SL2 cos()
If SL is isotropic: GB = 2SL cos()
As  >0 the liquid phase will spread over the whole GB. This
corresponds to "complete" or "perfect" wetting of the GB
Condition for complete wetting: GB = 2SL
Example of GB wetting
Since GB is more anisotropic than SL, there can be conditions
where some high energy GBs are completely wet while low energy
GBs are still dry.
Wet GBs will lead to "liquid metal embrittlement"
Wetting and prewetting transitions
T
liq
adsorption
The temperature above which a given GB is wet is known as its
wetting temperature (TW, shown by dashed red line on the PD below)
TW

//
X
X
X0
Consider what happens when two-phase coexistence is approached from
the single -phase domain:
At T>TW, there is a jump in adsorption at GB, known as a prewetting
transition. It is a precursor of the liquid film that will form when the Lphase becomes stable (the location is indicated by dotted green line)
At T<TW, no adsorption transition is present
Diffusion in Solids -- mathematical formalism
Diffusion represents flow of material, which occurs in order to
eliminate chemical potential gradients. In general, uniformity of
composition leads to elimination of chemical potential gradients
c''
c
t=0
t = t1
t = t2
t 2 > t1 > 0
c'
0
x
The flux J [m/A.t] at any time t and position x along the concentration
profile is given by Fick's first law:
c
J  D
x
where D [ l2/s] is the diffusion coefficient
Relationship between Fickian and chemical potential formalisms
d
J  M
dx
From the definition of chemical potential in the case of ideal or dilute
solutions, it is easily shown that:

or

d RT dc

dx
c dx
MRT dc
J 
c dx
So, for these simple solutions:

MRT
D
c
Back to Fick. The evolution of the profile with time and position is
given by Fick's second law:
c
J
 2c

D 2
t
x
x
For semi-infinite solids, the solutions of this PDE take the form:

 x 
c( x,t )  A B erf 

 4Dt 
where A and B are constants that may be determined from the
applicable initial and/or boundary conditions. For example, with:
 c(x,0) = c' for x < 0, and c(x,0) = c'' for x > 0
IC:
A = (c'' + c')/2; B = (c'' – c')/2
 x 
c  c 
c( x,t )  c 
1 erf 

2 
 4Dt 
Atomic mechanisms
Fraction of vacant sites:
 G f 
V
xV  e xp

 RT 




Rate of atom-vacancy exchange:
 G m 
V
   e xp

 RT 



The "self" diffusion coefficient is proportional to the product of the
vacancy fraction and the rate of atom vacancy exchange
Dself

 G f  G m
V
V

 xV    e xp 

RT




 S f  S m
V
  e xp V


R




  E f  E m
V
V
e xp
 
RT
 
This provides a hint as to the origins of the Arrhenius relation which
empirically describes diffusivity
 Q 
D  D0 e xp

RT







Some examples of the relative magnitudes of diffusivity
1200°C
900°C
400°C
600°C
10-10
10-12
D, m2/s
10-14
10-16
10-18
10-20
10-22
10-24
0.6
Fe in bcc Fe
Fe in fcc Fe
C in bcc Fe
C in fcc Fe
Mn in fcc Fe
0.8
1
1.2
1000/T(K)
1.4
1.6
1000°C
500°C
10-10
2
D, m /s
10-12
10-14
10-16
10-18
10-20
0.7
Ag in Ag
Ag in Ag GB
Cu in Cu
Zn in Cu
0.8
0.9
1
1.1
1000/T(K)
1.2
1.3
1.4
Summary
Interfacial thermodynamics, and Gibbs' treatment of adsorption
(segregation).
Complements to that treatment obtained from statistical thermo,
monolayer (and multilayer) models
Dependence of interfacial energy on adsorption
Anisotropy of interfacial energy (surfaces -- ECS, and GBs)
GB structure
Vector interfacial equilibrium (GB-surface, GB-second phase) and
wetting
Mathematical and physical descriptions of diffusion