Basic Elements of Thermodynamics of Surfaces •Short Reminder of Bulk Thermodynamics •Formation of Surfaces and Related Free Energy •Definition of Thermodynamic Variables Related to the Surface •Surface Strain and Stress Tensors •Gibbs Equation of Adsorption •Anisotropy of the Surface Free Energy (g-plot) •Vicinal Surfaces and Model for Their Surface Free Energy •Surface Free Energy and Crystal Shape (Wulff Theorem and Wulff Construction)
•Semi-infinite Crystals (Buckling and Faceting) •Finite Crystals (Crystallites)
•Roughening Transition •Line energy and two dimensional islands
Adhesion of water
molecules to tube surface
is stronger than cohesion
between water molecules.
Adhesion of mercury molecules to
tube surface is weaker than
cohesion between mercury
molecules.
The shape of the surface (meniskus in the case of liquids)
depends on the relative strength of adhesion and cohesion
Surface phenomena are driven by the minimization of
the surface free energy achieved
either:
1) by reducing the area of the surface by assuming a
spherical shape
Reconstructed fcc(110)
or
2) by altering the local
surface atomic geometry
reconstructing in a way
which reduces the surface
free energy Relaxed Surface (d1-2 < dbulk )
fcc(100)+c(2x2)-Ad
or
3) by adsorption
form the gas
phase
For more complicated situations, eg. alloys,
the surface energy may be reduced :
1. by segregation
2. by adsorbate induced
segregation
Increasing concentration
of “red element”
Before adsorption After adsorption
3. by forming a
superstructure
PtCo(100)
Pt – bright
Co – black (almost
invisible)
4. by clustering
U U(S,V, N)
Bulk Thermodynamics: A Short Reminder
Any one-component system, in equilibrium, is described completely by the
internal energy U
where S is the entropy, V the volume, and N the number of moles.
The infinitesimal variation of U is thereby
dU U
S
V ,N
dSU
V
S,N
dV U
N
S,V
dN
which becomes dU TdS pdV dN
with T the absolute temperature, p the pressure, and µ the chemical potential.
The extensive property U is described as U(S,V,N) U(S,V,N)
Reminding that NpVTSU
And combining its differential with the above equations one arrives at the Gibbs-
Duhem equation among the intensive variables
SdT Vdp Nd 0
The question is now: How does all this stuff change when treating a system
with a free surface?
When a surface of area A is created, via a cleavage process, the total internal
energy of the system must increase by an amount proportional to A, since
otherwise this process would occur spontaneously.
Bulk + Surface Thermodynamics
The Energy U is thus: U TS pV N gAwith the constant of proportionality g called surface tension.
Warning: The surface tension (g) can be regarded as an
excess free energy/unit area
The work may in general be of mechanical, chemical or electromagnetic origin
δW= δWmech + δWchem + δWelectr
At the surface we have to add a term corresponding to the formation
of an extra surface dA
with σij and εij the components of stress (N/m) and strain (pure number) tensors
which have 6 independent components
ji
ijijdA,
SOLID SURFACE/
INTERFACE
VAPOR
Definition of the interface region: At equilibrium, at any finite T and p, the semi-
infinite solid co-exists with its vapor and the system can be modeled as:
S S1 S2 Ss
V V1 V2 Vs
N N1 N2 Ns
Si siVi
Ni iVi
2,1i
After Gibbs, we ascribe definite amounts of the extensive variables to a
given area of surface and thus, calling the volume 1 and the vapor phase 2
where:
With si and ϱi entropy and particle density
Once the surface volume Vs is chosen, the other surface variables, Ss and Ns, are
defined as excesses
We then have the following relations:
Ss S1 S2
Vs V1 V2
Ns N1 N2
Warning: The boundary conditions are not unique!
However, it will result that one can always choose a subset of the surface
excesses with values independent of any specific conventional choice.
dzzzAN sS interface
2,1 )]()([
ij
jiNVSijAVS
ANSANV
dAU
AdNN
U
dVV
UdS
S
UdU
,,,,,
,,,,
/
Now, consider the effect of infinitesimal variations in the area of the system,
e.g. by stretching.
Assuming that linear elastic theory holds, one gets
dU TdS pdV dN A ijdijij
and thus
where ij and ij are the components of the surface stress and surface strain
tensors, respectively.
Warning: be aware of the dimensions of ij (Force/unit length) and ij (pure
number)
Taking into account that dA A dij iji, j
One arrives at the Gibbs-Duhem equation for the total system
0)(,
ji
ijijij dANdVdpSdTAd gg
0)(,
ji
ijijijsss dAdNdpVdTSAd gg
However, the original Gibbs-Duhem equation
holds still for each of the two bulk phases separately.
Applying it twice, one arrives at
which is the Gibbs adsorption equation relating surface excess variables.
The number of independent variables is only three because of the Gibbs
Duhem relationships in solid and vapor phases.
Writing dµ and dp in terms of dT using the bulk phases Gibbs Duhem
equations, we get
0)()(
,12
21
12
1221
ji
ijijijsss dAdTss
Nss
VSAd g
g
The interesting thing is that one can now show that the quantity in brackets
is independent of the arbitrary boundary conditions which define Ns, Vs,
and Ss.
SdT Vdp Nd 0
0sV
Consequently, after Gibbs, we can choose
Gibbs assumed for the surface volume that the number of particles outside the surface
plane is equal to those missing in the inside. This is known as the equal area
convention. However, one can assume and
with no loss of generality.
0)(,
, ji
jiijijs dAdTSAd gg
Ss Ag
T
ij g ij g
ij T
Warning: surface tension (g) and surface stress () are in general not identical
(except for liquids) . Surface tension and its derivative are of the same magnitude.
If ij can vanish or become negative. A negative stress implies a
reduction of the energy when new surface is created.
and obtain:
0sN
for the surface
entropy
for the
surface stress
g 0
For Au(111) g 0
and Au foils contract (so called creep) when atom diffusion is activated (e.g. by
heating a gold foil near the melting point) because atomic diffusion occurs under
the influence of surface forces.
The surface tension g can be measured by opposing the creep with known external
forces.
An estimate of surface tension can be obtained from
g Ecoh Zs Z Ns
where Ecoh is the bulk cohesive energy, (Zs/Z) the fractional number of
bonds broken (per surface atom), and Ns the surface areal density.
Using typical values:
Ecoh ≈ 3eV, (Zs/Z) ≈ 0.25, Ns ≈ 1015 atoms/cm2
we get g ≈ 1,200 erg/cm2
Since this stress takes place over the surface thickness (1nm) the
corresponding pressure is 1 Gpa. This means that neglecting the external
pressure in surface thermodynamics is in general justified.
Anisotropy of g
The surface tension of a planar solid depends on the crystallographic
orientation of the sample.
Vicinal surfaces: surfaces slightly misaligned with respect to a
specific direction
[1n0]
d) q
d
The anisotropy of the surface tension is represented
via the g-plot constructed by drawing a vector from the
origin in the direction n (defined by its polar and
azimuthal angles q and f) with a length equal to the
surface tension, g(n), of the surface plane perpendicular
to n.
The asphericity of the g-plot reflects the anisotropy of g which has minima in
the directions n0 corresponding to close-packed surfaces.
For a vicinal surface, showing a periodic succession of terraces and steps,
with b the energy per unit length of a step, we get:
g (n) g (n0 )bq
dwhere n0 defines a close-packed surface, q is the angle between n and n0, and d is
the interplanar distance along n0. |q|/d is the density of steps.
Lev Landau (1965) showed that g(q)
has a cusp at every angle of a
rational Miller index.
The sharpness of the cusp is a
rapidly decreasing function of index:
dg
dq
1
n4
For large q values, the density of steps increases and one has to include the
energy of interaction between steps.
dg/dq has discontinuities at q = 0, more precisely:
dg
dq
q0
2 b d
and the g-plot shows cusps in directions typical of the most close-packed surfaces.
dSFS
s )(ng
Finite Crystal limited by a surface S. The equilibrium shape must minimize the excess surface free energy
while preserving the volume:
The variational geometric problem was solved by Wulff (1901).
•Draw a radius vector intersecting the polar plot at one point and making a fixed angle
with the horizontal.
•Construct the plane perpendicular to the vector at the intersection.
•Repeat this procedure for all angles.
The interior envelope of the resulting family of
planes is a convex figure whose shape is that of the
equilibrium crystal.
Wulff construction (more precisely)
Let’s introduce a surface tension γp(θ) = γ(θ)/lp (defined with respect to
the length scale projected onto the surface lp=lcos θ) with respect to the
angle θ.
dp
qbg
q
qgqg
tan
)cos(
)()( 0
β± is thereby the line tension for up and down steps (not identical for (111)
surfaces where A and B steps have different structures). The expression can be
considered as the first order of a series expansion
...)( 2
210 ppp gggqg
in which the higher order terms correspond to the step - step interactions
(proportional to 1/L for the γ2p2 term).
Semi-infinite Crystal limited by a plane S with normal at q =0
Now let us study its stability relative to a small polar buckling preserving
the average orientation.
The free energy of the buckled S’ surface is
SS
S
dAdAF
qqgqg
cos)(')(
dAd
ddA
d
dAF
SS
s
0
2
22
0
)(2
1)0(
qgq
gq
q
gqg
An expansion up to second order in q gives:
The second term vanishes for symmetry reasons and the energy involved
in the deformation is thus the last term.
g (0) (d2g dq 2
)q0 0
the flat surface is stable (or metastable)
g (0) (d2g dq 2
)q0 0
the flat surface is unstable and will minimize its
energy by developing facets
Facetting
g1 g2
g (0) (d2g dq 2
)q0 0
Imagine to have a planar surface with a large surface free energy in a highly
anisotropic crystal.
Some energy can be gained by replacing the smooth surface with a saw-tooth
profile while preserving the average orientation.
Whenever the surface stiffness is negative the surface facets generating
more stable nanosized surface areas (as e.g. for surface reconstruction)
surface stiffness
Crystal temperature dependence of γ
The Helmholz free energy F=U-TS
decreases with T and so does the surface
tension. The cusps become less and less
well defined. When γ becomes isotropic
the surface is said to be rough and the
crystal assumes a spherical shape
Construction
of the Wulff
equilibrium
shape
Strongly anisotropic case
Weakly anisotropic case
Herring construction to
determine the saw-tooth
profile typical of facetting
bcc crystallite
Tensile and compressive stress at bare
and adsorbate covered surfaces
σ>0 σ<0
The charge which is not shared by the missing atoms redistributes on the
surface plane causing an attractive force on the ion cores. A free standing 2D
layer of atoms would thus have a smaller lattice constant. Since teh atoms are
kept in register by the substrate lattice a tensile tension of approx 1N/m builds
up (corresponding to a bulk stress of 1 Gpa)
Consequencies for crystal growth Warning growth occurs in conditions
far from equilibrium
For an anisotropic solid the condition of
equilibrium of i different phases is:
For a deposit, and assuming a rough surface in order to neglect the derivative of γ
no α value satisfies the above
condition : case of complete
wetting and pseudomorphic growth
γi interface, γdep deposit and γs
substrate tension
otherwise
Roughening Transition
At T = 0K a stretched line (or surface) is straight (or flat) on a
microscopic scale.
When T increases, thermal fluctuations appear:
The line becomes sinuous and the surface buckles.
Warning: One must now include explicitly also entropy effects when
treating these T-dependent phenomena
(a) Only a few thermally
excited defects are present
(b) Long wavelength
variations in height
Thermal fluctuations: Root mean square deviation of the
position with respect to the average position of an infinite
line or surface.
Thermal fluctuations may remain finite or diverge
Finite fluctuations --> Smooth line or surface
Diverging fluctuations --> Rough line or surface
Surfaces can be sorted out into two families:
Surfaces where roughness exists at any T 0K
Surfaces where roughness exists above a critical
temperature TR (roughening transition temperature)
passing through a phase transition
Burton and Cabrera (1949) suggested the possibility of the
roughening phase transition at surfaces
Theoretical Approach
1) The system is treated as a continuum, the effect of the atomic
structure (lattice potential) being introduced via a pinning potential
favoring given periodic positions of the line or surface.
1a) In the absence of this potential the line and the surface are
rough at any T≠0K
1b) When this potential is taken into account the line remains
always rough while there exists a roughening transition for the surface
2) The discrete atomic structure is explicitly taken into account ab
initio.
The line is always rough while the surface exhibits a roughening
transition depending on its detailed structure.
Warning: The roughening transition can be actually observed
only if TR < Tm where Tm is the melting temperature
One of the simplest models is the so-called solid-on-solid (SOS)
model
The crystal is viewed
as a stacking of
elementary cubes.
As T increases
fluctuations appear
along with more and
more defects. 1) The cubes are arranged into columns of different heights, hi
2) These columns (one for each surface atom) can interact each other
3) J represents the finite energy cost if nearest neighbor columns differ in
height by one lattice constant
4) In general one takes
H J hi hj
i, j
2
under the condition that the surface is perfectly flat at T = 0K
SOS Model: The lowest energy excitations are monoatomic steps coalescing into plateaus
1 1 2
0
z = 0
The number of possible loops of this length is equivalent to the number of self-
avoiding random walks returning to the origin in L/d steps.
The number of these loops is zL/d
The contribution to free energy is thereby )ln( zkTJd
LTSUF
A loop of length L bounding a plateau
has energy (JL)/d where d is the
lattice constant.
Below the roughening transition temperature, kTR = J/ln z
the contribution to F is positive and L = 0 is favored.
Above TR, loops of arbitrarily large length occur and the surface
becomes rough.
Equilibrium shapes of Pb crystals at selected T’s
Morphology of Pb
crystals as a function
of the growth
temperature T
Facets
T≥393 K
323 K ≤T ≤393 K
T≤323 K
The hierarchy of
equilibria
The islands are in equilibrium
while the surface is not and
evolves with time. This is
connected to atom diffusion
which takes place over very
different time scales along the
border of the islands and across
the flat surface between islands
Island shape : The Wulff construction for a 2 dim island should in
principle show straight lines. This is however not the case because there is a
theorem stating that there are no phase transitions in one dimensional
systems at finite temperature for interactions decaying faster than 1/x2 .
In other words fluctuations of one dim systems are too large.
Noteworth exceptions are the reconstructed Au(111), Au(100), Ir(100) and Pt(100).
The borders are then stabilized when the reconstruction matches the terrace width
making the system effectively 2 dim.
The surface reconstruction of Au is lifted in contact with an
electrolyte. In such cases the shape of the islands behaves
differently than in ultra high vacuum
Surface tension and
thin film growth