Thermodynamics of surface and interfaces – (Gibbs 1876 -78)

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 o

f the surface of area A, the area change dA is a product of that portion of peri

meter and the length moved.

dAdNpdVTdSdU ii

i

Work term - dA; force x distance, and appear in the combined 1st an

d 2nd laws of thermodynamics as

J. W. Gibbs, collected works, Yale Univ. Press, New Haven, vol.1(1957), p. 219 ~ 331.

For a system containing a plane surface this equation can be reading integrated

:

i ii

U TS PV N A

Strictly speaking , is defined as the change in internal energy when

the area is reversibly increased at constant S, V and Ni (i.e., closed

system).

i

ii NPVTSU

A 1

where U – TS + PV is the Gibbs free energy of the system.

and rearranging for yields.

And ii

i N is the Gibbs free energy of the materials comprising the

system.

def Surface Excess Quantities

Macroscopic extensive properties of an interface separating bulk phases a

re defined as a surface excess.

Thus is an excess free energy due to the presence of the surface.

There is a hypothetical 2D “dividing surface” defined for which the para

meters 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 prese

nt in the same volume if the phases were homogeneous right up to the

“ Dividing Surface ” i.e.,

xxxx totals

The real value of x in the system

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 = 2An

Then letting A = a2 where a lattice spacing 22a

where n/A is the # of broken bonds / unit-area and the is the energy per bond i.e., the well depth in the pair-potential.

pair potential

r

U(r)

AddAdU

and dU df AdA dA

If the solid is sketched such that the surface area is altered

the energy ddAAA

daaa

The total energy of the surface is changed by an amount..surfS AU

Surface Stress and Surface Energy

Then in general the relationship between surface stress and surfaceenergy is given by,

1,2 j i, fij

ijij

For a surface with 3-fold or higher rotational symmetry fij is isotropicand the surface stress can be treated as a scalar.

dAdAf

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.

dAdAf

dAdU for an isotropic surface

For an anisotropic surface f is a tensor quantity and

ijijijf

where double sum in the strains and .)1(0 ijAA ijA

dA

The anisotropy of surface energy

Surface energy is a function of orientation – crystalline solid. For a liquid γ i

s isotropic and the equilibrium shape minimizes the surface / volume ratio. Fo

r example, the equilibrium hape of a soap bubble is a sphere.

Experimentally it has been found that cuts of crystals off a low index orient

ation 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 al

ong a low index direction owing to step structure.

2

sincos2

a

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.

In a nn. central force model, for surfaces forming a stepped structure

is given by:

(10)“cusps” @ 0

Polar Plot of

Low-index plane have cusps in plots at 0 K which tend to get ro

unded off at higher temperatures.

Inner envelope of normals defining the equilibrium of shape of crystal.

Equilibrium shape of a crystal obtained when minimumplanes

ii A

and this is given by the Wolff’s theorem.

( 0 1 )( 1 1 )

( 1 0 )

Example of a 2D polar plot of

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.

( 0 1 ) ( 1 1 )

( 1 0 )

Einstein mobility relation

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.

)11()01()11()01()11()01( ; ; vvFF

Faster growth

Slower growth

Faster growth

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.

Estimation of interfacial energies

Recall: 2/aZ

Types of Interfaces

(d) Solid / solid

chemical

structural

(a) solid / vapor

(b) liquid / vapor

(c) solid / liquid

where is the bond energy, Z is the number of near neighbors

and is the atom density of surface.

a

For the liquid / vapor interface -

vapaLv LZ 2/

For the solid / liquid interface – entropic effects dominate.

/ 2SL a fusionZ S T

m

fusfusion T

LS

Note the temperature dependence.

For many situations, these values provide reasonable estimates.

For the solid / vapor interface -

fusionvap LLL nsublimatio

subasv LZ 2/

( J / m2 ) near Tm

( J / m2 ) @ 25ºC

Sn 0.68 MgO 1.0

Ag 1.12CaF2 (111)

0.45

Pt 2.28CaCo3 (1010)

0.23

Cu 1.72LiF

(100)0.34

Au 1.39NaCl(100)

0.30

T (°C) ( 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

SV SV LV

Note that near Tm, lv ~ sv.

Values of SV Values of LV

( J / m2 ) near Tm

Al 0.093

Cu 0.177

Fe 0.204

Pb 0.033C2H2(CN)2 [succinonitrile] 0.009

Nylon 0.020

vdPsdTd p

sT

and for metals 3 210 / weak T dependences J m K

Values of (inferred from nucleation exp)SL

SL

Temperature dependence of ( solid / vapor, liquid / vapor ) recall that for a 1 component system:

Solid / Solid Interfaces

(i) Chemical bonding(ii) Structural bonding ( say phases have different crystal structure)

structuralchemical

y (distance)

xα

xβ

xα is the mole fraction of A in α

xβ is the mole fraction of A in β

Chemical contribution to :chemical

Consider a general inter-phase α/ β boundary. can be thought of as being composed of 2 terms :

Let U(x1, x2) be the sum of the bond energy per unit area between

planes of composition x1 and x2

1, 2 / 2 1 1 1 1 a AA BB ABU x x Z x x V x x V x x x x V

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:

2, 1/ 2 , , / 2 i aU U x x U x x U x x x x

where 2/1 BBAAAB VVVZ

Ω can be estimated from Ω/2R = T critical (see Reg. Sol. Theory)

Regular solution model of an interfaces (Becker 1938)

For metal / metal interfaces :

219 /10 2/ ma atJatomeV /10 / 1.0 ~ -20

For close to pure metal interfaces :

0 ,1 xx

so 21 /10~ mJU i

Typical values of lattice matched (coherent) interfaces energies range from 10-3 ~ 10-1 J/m2.

* The diffuse chemical interface (1-D estimates)

– variation in the mole fraction of A atom 1 plane to the nextx

interplaner spacinga

y (distance)a

x

x

x

2 2/ xxU ai generalize to a continuum

2

1,22

i

iiai a

xxΩa/ρU

, 1i ix x dx a dy

2

22

12i adx dx

dyU ρ a

dyK/ Ω

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 oftwo 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 comp

ositional gradient K(C)2

The total free energy is expressed as a functional ( a function of afunction) i.e.,

20 ( ) ( )

VG g c K C dV

for metals K 10-19 J/m

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.

Structural Interfacial energy: ( to be discussed in more detail later, see coherence)

Def misfita a

ma

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.

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’)

Grain boundaries

Cut ABCD

Consider the triple point of a gb junction:

Grain 1

Grain 2

Grain 312

23

13

12

23

13

Rot @ y-axis

Twist boundaryTilt boundary

Rot @ x-axis

Ay

z

x

B C

D

Herring (1951) showed that by balancing the forces for a virtual change in the orientation of the triple junction :

1313

1323

23

232313132312 sinsincoscos

where the

i

i

terms are called “ surface torque” terms.

For high angle gb the torque terms can be neglected and

23

23

13

13

12

12

sinsinsin

d23

d13

13

23

12

Wetting (Contact) Angle

LV

solid

SVSL

Force Balance

cosLVSLSV

Def

s SL SV

s

LV

cos

is called the wetting or contact angle.

s SL SV s

LV

cos

θ = 0 Complete wettings LV

θ < 90º0s

θ > 90º0s

θ > 180ºs LV No wetting