Thermodynamics of surface and interfaces – (Gibbs 1876 -78)
J. W. Gibbs, collected works, Yale Univ. Press, New Haven, vol.1(1957), p. 219 ~ 331.
Define
 :
Consider
 to be a force / unit length of surface perimeter.
(fluid systems)
If a portion of the perimeter moves an infinitesimal of distance in the plane
of the surface of area A, the area change dA is a product of that portion of
perimeter and the length moved.
Work term -
 dA; force x distance, and appear in the combined 1st
and 2nd laws of thermodynamics as
dU  TdS  pdV   i dN i   dA
i
 is defined as the change in internal energy when
Strictly speaking ,
the area is reversibly increased at constant S, V and Ni (i.e., closed
system).
For a system containing a plane surface this equation can be reading integrated :
U  TS  PV   i Ni   A
i
and rearranging for

yields.

 1 
   U  TS  PV   i N i 
 A 
i

where U – TS + PV is the Gibbs free energy of the system.
And
 N
i
i
i
is the Gibbs free energy of the materials comprising the
system.
Thus is an excess free energy due to the presence of the surface.
def
Surface Excess Quantities
Macroscopic extensive properties of an interface separating bulk phases
are defined as a surface excess.
There is a hypothetical 2D “dividing surface” defined for which the
parameters of the bulk phases change discontinuously at the dividing
surface.
def
The excess is defined as the difference between the actual value of the
extensive quantity in the system and that which would have been
present in the same volume if the phases were homogeneous right up to
the “ Dividing Surface ” i.e.,
x x
s
total
The real value of x in
the system


 x x


The values of x in the homogeneous
 and phases
Solid and liquid Surfaces
In a nn pair potential model of a solid, the surface free energy can be thought
of as the energy/ unit -area associated with bond breaking. :
work/ unit area to create new surface =
where n/A is the # of broken bonds / unit-area and the
bond i.e., the well depth in the pair-potential.
Then letting A =
a2
where a lattice spacing

n
  2
A

is the energy per

2a 2
pair potential
If the solid is sketched such that
U(r)
the surface area is altered
a  a  da
r
the energy

The total energy of the surface
U S  Asurf .is changed by an amount.
dU  dA  Ad
and
A  A  dA
    d
dU
d
 f   A
dA
dA
Surface Stress and Surface Energy
Then in general the relationship between surface stress and surface
energy is given by,
  

f ij  ij   
 ij 


i, j  1,2
For a surface with 3-fold or higher rotational symmetry fij is isotropic
and the surface stress can be treated as a scalar.
d
f  A
dA
def
f - The surface stress is the reversible work/unit area associated with the
creation of new surface while altering its density by elastic stretching or
compressing.
dU
d
 f   A
dA
dA
for an isotropic surface
For an anisotropic surface f is a tensor quantity and

f ij   ij 
 ij
where
A  A0 (1   ij ) double sum in the strains and dA   ij
A
.
The anisotropy of surface energy
Surface energy is a function of orientation – crystalline solid. For a liquid γ is
isotropic and the equilibrium shape minimizes the surface / volume ratio. For
example, the equilibrium hape of a soap bubble is a sphere.
Experimentally it has been found that cuts of crystals off a low index
orientation equilibrate to form stepped structures such that the steps are
composed of low-index surfaces.
broken bond

( 1 0 ) plane
Bond density along direction defined by θ is greater than the bond density
along a low index direction owing to step structure.
In a nn. central force model,
 for surfaces forming a stepped structure
is given by:
cos  sin   
a2
2
where a (lattice parameter) is the unit of length and ε/2 the bond energy,
θ is the mis-orientation with respect to a low-index plane.
Polar Plot of

“cusps” @

  0
(10)
Low-index plane have cusps in    plots at 0 K which tend to get
rounded off at higher temperatures.
Example of a 2D polar plot of

(01)
(11)
(10)
Inner envelope of normals defining
the equilibrium of shape of crystal.
Equilibrium shape of a crystal obtained when
  A  minimum
i
planes
and this is given by the Wolff’s theorem.
i
Wulff construction
Wulff’s theorem : The equilibrium shape is obtained by taking the inner
envelope of the normals. This envelope defines a shape geometrically
similar to the equilibrium shape of the crystal
Any surface which does not appear in the equilibrium shape can lower
its energy by forming a stepped structure, composed of planes which do
appear in the equilibrium shape.
Often actual morphologies are determined by kinetic considerations.
Suppose the velocity of the interface is controlled by surface diffusion.
(01)
(11)
Einstein mobility relation
(10)
  MF
If the temp. dep M is about the same for all the orientation v is determined
by F. Generally the lower
 , the higher F so: e.g.
 ( 01)   (11)
;
F( 01)  F(11)
Faster growth
Slower growth
Faster growth
;
v( 01)  v(11)
The surfaces which grow faster tend to shrink in size.
Growth of crystals occur by a ledge process.
Fast
Slow
Faster faces grow out; overall growth tends to be limited by slowest faces.
Types of Interfaces
(a) solid / vapor
(b) liquid / vapor
(c) solid / liquid
chemical
(d) Solid / solid
structural
Estimation of interfacial energies
Recall:
where

  Z  a / 2
is the bond energy, Z is the number of near neighbors
and  a is the atom density of surface.
For the solid / vapor interface -
  Lsublimation  Lvap  L fusion
 sv  Z  a / 2Lsub
For the liquid / vapor interface -
 Lv  Z  a / 2 Lvap
For the solid / liquid interface – entropic effects dominate.
 SL  Z  a / 2 S fusion  T
Note the temperature dependence.
 L fus 

S fusion  
 Tm 
For many situations, these values provide reasonable estimates.
Values of  SV
 SV
 SV
( J / m2 )
near Tm
Sn
Values of  LV
( J / m2 )
@ 25ºC
0.68
MgO
Ag
1.12
CaF2
(111)
0.45
Pt
2.28
CaCo3
(1010)
0.23
1.72
LiF
(100)
0.34
1.39
NaCl
(100)
Cu
Au
Note that near Tm, lv ~ sv.
1.0
0.30
T (°C)
 LV
( J / m2 )
H2O
25
0.072
Pb
350
0.442
Cu
1120
1.270
Ag
1000
0.920
Pt
1770
1.865
NaPO
620
0.209
FeO
1420
0.585
Al2O3
2080
0.700
Values of  SL (inferred from nucleation exp)
 SL ( J / m2 ) near Tm
Al
0.093
Cu
0.177
Fe
0.204
Pb
0.033
C2H2(CN)2 [succinonitrile]
0.009
Nylon
0.020
Temperature dependence of  ( solid / vapor, liquid / vapor ) recall that
for a 1 component system:
d   sdT  vdP
  

  s
 T  p
and for metals s  103 J / m2 K  weak T dependence
Solid / Solid Interfaces
Consider a general inter-phase α/ β boundary.   can be thought of as being
composed of 2 terms :
(i) Chemical bonding
(ii) Structural bonding ( say phases have different crystal structure)
   chemical   structural
Chemical contribution to
 chemical
:
xα
xα is the mole fraction of A in α
xβ
xβ is the mole fraction of A in β
y (distance)
Regular solution model of an interfaces (Becker 1938)
Let U(x1, x2) be the sum of the bond energy per unit area between
planes of composition x1 and x2

U  x1, x2   Z  a / 2  x xVAA  1  x  1  x  VBB   x 1  x   1  x  x  VAB
where VAA, VBB, and VAB are the bond energy.
In analogy with the regular solution model we define an excess energy,
Ui, due to the interface:
U i  U  x , x   1/ 2 U  x , x   U  x , x      a / 2    x  x 
where
  Z VAB 1/ 2VAA  VBB 
Ω can be estimated from Ω/2R = T critical (see Reg. Sol. Theory)
2

For metal / metal interfaces :
 a / 2  1019 / m 2  ~ 0.1 eV / atom  10-20 J / at
For close to pure metal interfaces :
x  1,
so
x  0
U i   ~ 10 1 J / m 2
Typical values of lattice matched (coherent) interfaces energies range from 10-3 ~
10-1 J/m2.
* The diffuse chemical interface (1-D estimates)
x
x
x
y (distance)
a
a
 interplaner spacing
x – variation in the mole fraction of A atom 1 plane to the next
U i   a / 2  x  x 
 generalize to a continuum
2
 xi  x ,i1 
2
U i   ρa / 2Ωa  

i
a


 x
i
 x ,i 1 
 dx
2
a  dy
2
 dx 
 dx 
U i   ρa / 2  Ωa    K1  
 dy 
 dy 
2
2
composition gradient
K1  Gradient energy coefficient in 1D
* important in spinodal decomposition
Cahn – Hilliard Free Energy (1958)
Consider a small region of material with a chemical inhomogeneity. :
The free energy per unit volume can be thought of as being composed of
two terms:
(i) g0(C) homogeneous free energy per unit volume, if the material was of
homogeneous composition ( Regular sol. Model)
(ii) g i(C) inhomogeneous free energy owing to the presence of the
compositional gradient  K(C)2
The total free energy is expressed as a functional ( a function of a
function) i.e.,
G    g0 (c)  K (C )2  dV
V
for metals K  10-19 J/m
Structural Interfacial energy:
( to be discussed in more detail later, see coherence)
Classification of structural interfaces:
(i) Coherent
lattice matched systems
some x’tal structure or
( 1 1 1 )fcc / ( 0 1 1 )hcp
or ( 1 1 1 )fcc / ( 1 1 0 )bcc and etc.
(i i) Incoherent




The structural misfit energy is most easily accommodated by forming
“misfit dislocation” in the interface.
Def
misfit
m
a  a
a
For “large” m misfit dislocations form.
Grain boundaries, twin boundaries, stacking faults are examples of
structural interfaces which can have ( 1 component ) no chemical term.
Grain boundaries
Gbs are incoherent interfaces defined by the relative misoreintation between
grains.
To specify a gb define the orientation of the crystallites with respect to one
another and the orientation of the boundary with respect to one of the crystallites.
In 3D the specification of 3 angles ( with respect to the coordinate axes) is
necessary to describe the relative orientation between crystals and 2 angles
specify the boundary orientation with respect to one of the crystal axes. ( see
Bollman, 1970, “ crystal defects and interfaces’)
Cut ABCD
z
Rot @ x-axis
B
Rot @ y-axis
C
y
D
A
Tilt boundary
x
Twist boundary
Consider the triple point of a gb junction:
13
Grain 3
Grain 1
12
23
23
13
12
Grain 2
Herring (1951) showed that by balancing the forces for a virtual change in
the orientation of the triple junction :
 13
d23
 12
 23
d13
 23
 13
 12   23 cos13   13 cos 23 
sin  23 
sin 13
 23
13
  
where the  i  terms are called “ surface torque” terms.
  i 
For high angle gb the torque terms can be neglected and
 13
 23
 12


sin 12 sin 13 sin  23
Wetting (Contact) Angle
Force Balance
LV
SL
SV

solid
 SV   SL   LV cos
Def
s   SL   SV
s
cos   
 LV
 is called the wetting or contact angle.
s   SL   SV
 s   LV
θ=0
s  0
θ < 90º
s  0
θ > 90º
 s   LV
θ > 180º
s
cos   
 LV
Complete wetting
No wetting