Basic Elements of Thermodynamics of Surfacesrocca/Didattica/Surface Science and...Basic Elements of Thermodynamics of Surfaces •Short Reminder of Bulk Thermodynamics •Formation

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(

qq

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