Surface Energy, Surface Energy, Surface Tension & Shape of CrystalsSurface Tension & Shape of Crystals

MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING ENGINEERING

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh

AN INTRODUCTORY E-BOOKAN INTRODUCTORY E-BOOK

Part of

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htm

A Learner’s GuideA Learner’s GuideA Learner’s GuideA Learner’s Guide

Let us start with a few observations: Crystals (which are ‘well grown’) have facets Under certain conditions of growth we may observe tree like patterns – known as

dendritic growth Two kinds of shapes of crystals are important: (i) growth shape and (ii) equilibrium shape Surface/interface energy plays an important role in determining the shape of a crystal.

Shape of Crystals

Note the facets

Video: Dendritic growth of crystal from melt

Video: Dendritic growth of crystal from melt

KDP crystals grown from solution

Electrodeposited nanocrystalline Al-Mg alloy powders(Photo courtesy: Dr. Sankarasarma Tatiparti)

Dendritic growth of crystals

Close to equilibrium shape

A cut through an infinite crystal creates two surfaces. The joining of two phases creates an interface.

(Two orientations of the same crystalline phase joined in different orientation also creates an interface called a grain boundary).

What is a surface and what is an interface?

Creation of a

surface

Creation of an

interface

Note: Surface can also be thought of a vacuum-material interface (or even a air-material interface)

(or materials)

(or materials)

Join

Cut and Separate

Consider the following dialogue:Kantesh: I suffered a loss of 4 crore rupees!Anandh: How did that happen?Kantesh: Last year I got a profit of 14 crores and this year I got a profit of only 10 crores- that is a loss of 4 crores!!

Did Kantesh really suffer a loss?!!

How to understand surface energy?

The accounting leading to the concept of surface energy is similar to the dialogue above (‘in some sense’).

To understand this further let us do the following ideal thought experiment:(i) start with atoms far apart (upcoming figure)

→ such that there is no ‘bonding’ (interactions) between them(ii) bring the atoms close to form a ‘bonded state’ with a surface

Let the energy of the ‘unbonded state’ be zero. Let the energy lowering on bond formation be Eb per

bond.

Each bulk atom is bonded to 4 atoms (as in the upcoming figure)

Energy lowering of bulk atoms = 4Eb → this is negative energy w.r.t to the unbonded state

Each surface atom is bonded to 3 other atoms onlyEnergy lowering of bulk atoms = 3Eb → this is also negative energy w.r.t to unbonded state!

We use a crystal to understand the concept

Cotd...

Hence, we have seen that surface energy is ‘not really an energy’ in the truest sense → it is a correction coming about because we had over counted the number of ‘fully bonded’ atoms. (Sir Richard Feynman may say that all forms of energy are accountant’s book keeping terms).

However, the effects of surface energy is very real and it is nice to hang on to the concept!

Energy lowering on the formation of infinite crystal/unit volume =

[ (number of atoms) 4Eb]

Energy of a crystal with a free surface/unit volume =[ (number of atoms) 4Eb] + [(number of surface atoms) 1Eb]

The reference state for the surface energy is the bonded state and not the

free state

An alternate calculation without invoking surface energy

Energy of a crystal with a free surface/unit volume =[ (number of bulk atoms) 4Eb] [(number of surface atoms) 3Eb]

This is the surface energy!

Schematic not to scale

Funda Check What is a broken bond?

The electron distribution in a material can be viewed in a simplified manner using the language of bonds. I.e. isolated atoms have a higher energy as compared to the atoms in a solid (we restrict ourselves to solids for now) and this lowering of energy can be visualized as a bond.

The lowering of energy can be reported as bond energy/bond.

The number and types of bonds an atom forms in the solid state depends on: broadly speaking the electronic configuration of the constituent atoms

Atoms on the surface have a lower coordination number as compared to atoms in the bulk of the solid. The missing ‘coordination’ can be viewed as a broken bond.

The surface need not be a mere ‘termination of the bulk’ and may undergo relaxation or reconstruction to lower its energy.

Also the surface may be considered a few atomic layers thick (i.e. it need not just be a monolayer of atoms).

Surface Energy and Surface Tension are concepts associated with liquids and solids. If the Gibbs Free Energy (G) of the solid or liquid is lower than a given gaseous state

under certain thermodynamic parameters (wherein the atoms are far apart without any interatomic forces), then the gas will condense (and form a solid or liquid).

The lowering in the Gibbs Free Energy is due to the cohesive forces in the liquid or the bonding forces in the solid.

The lowering in energy is calculated for an atom (or entity) fully bonded. The atoms on the surface are not fully bonded. The atoms on the surface have a higher energy than the bulk atoms (in the regime where

the solid or the liquid have a lower energy than the gaseous state). Hence the reference state for the surface is the bulk and not the gaseous state.

Surface Energy

Partly bonded surface atoms

When the calculation of the lowering of the energy of the system on the formation of the condensed state was done all the atoms were taken into account (assumed to be bulk atoms) → i.e. an over-counting was done

The ‘higher energy’ of the surface is with respect to the bulk and not with respect to the gaseous (non-interacting) state

Hence the reference state for the surface is the bulk and not the gaseous state

Hence, it costs energy to put an atom on the surface as compared to the bulk → origin of Surface Energy ()

The surface wants to minimize its area (wants to shrink) → origin of Surface Tension ()

LIQUIDS Surface Energy Surface Tension

SOLIDS Surface Energy Surface TensionExcept in certain circumstances

+ Surface energy is Anisotropic

[ ]

[ ]

Force F N

Length L m

2 2

[ ] [ ] [ ]

[ ] [ ] [ ]

Energy E J Nm N

Area A m m m

Dimensionally and are identical → Physically they are different type of quantities is a scalar while is a second order tensor

Let us look at the units of these two quantities

LIQUID SURFACE Characterized by one number → the surface density

SOLID SURFACE Has a structure and hence more numbers may be neededto characterize a solid surface

Crystalline surfaces → all the lattice constants will be required Amorphous surfaces → Density + a Short Range Order parameter

Surface Energy Surface Tension

Surface Energy Surface Stress (Tensor) Surface Torque

Liquids cannot support shear stresses (hence use of the term surface tension)

In the case of solids the term surface tension (which actually should be avoided) refers to surface stresses

A comparison of the solid and liquid surfaces

Surface Energy () → is the reversible work required to create an unit area of surface(at constant V, T & i)

Surface Tension () → is the average of surface stresses in two mutually perpendicular directions

The definition of surface tension in 2D is analogous to the definition of hydrostatic pressure in 3D

2x y

Surface stress at any point on the surface is the force acting across any line on the surface which passes through this point in the limit the length of the line goes to zero

Liquid surfaces are characterized by a single parameter: the density (atoms / area)

The short range order in liquids (including their surfaces) is spatio-temporally varying → hence no structure (and no other characteristic) can be assigned to the surface

Crystalline solids have a definite structure in 3D and hence additional parameters are required to characterize them

The order at the surface of a crystal can be different from the bulk

Amorphous solids have short-range order, but NO long-range order. Under low temperature conditions and short times (i.e. low atomic mobility regimes) the atomic (entity) positions are temporally fixed

Funda Check What leads to an increased interface energy?

We will try to make heuristic arguments to understand interface energy. As we have already noted surface is a special kind of interface between material and

vacuum/air. If the material on the two sides are ‘similar’, then the interface energy is low.

More the difference in the nature of the two materials more will be the interface energy. Similarity can be based on: (i) atomic structure (including crystal structure, mismatch in

atomic planes, etc.), (ii) bonding nature (including valence electron concentration), (iii) electronegativity difference etc.

Low energy interface if: Same crystal structure on both sides of the interface, Interface is coherent (continuation of atomic planes from one side to another), Similar bonding (say metals on both sides with similar valence electron concentration) or in more general terms similar electromagnetic structure, Atoms with similar electronegativity on both sides, etc. (The orientation of the crystals and interface also plays an important role).

High energy of interface if: Bonding is different, Crystal structure is different, Interface is incoherent.

We have focused on interface between crystalline materials above. Interfaces can be between amorphous and crystalline, crystalline and quasicrystalline etc.

Some more mathematical looking concepts!

Some readers may want to skip the pages with ‘too much math’ and get to pages of interest.

Surface/Interface Effects become important

Interface has large curvature

(Surface : Volume) is large

When surface effects are important it is not advantageous to use pressure to characterize the state of the system → as pressure is different across a curved interface

T and (Chemical potential) are have the same value across the system and should be used to describe the state of the system

Interface

The thickness of the interface layer is determined from the equilibrium constraint that the chemical potential of each species present is constant throughout the system

Any variation in chemical potential will tend to lead to mass transport

12

Variation of thermodynamic function across the interface

2

[ ( ) ]v

dcF An f c k dx

dx

F → Helmholtz free energy f(c) → ‘F’ per molecule of a homogenous system of composition ‘c’ nv → No. of atoms per unit volume

A → Cross sectional area k → Constant for small gradients

Gradient term:Contribution due to variation

in composition

Geometrical dividing surface

Instead of the diffuse interface a geometrical dividing surface can be used if: the radius of curvature >> thickness of the transition layer (or dimension of crystal)

The dividing layer is positioned within the transition layer such that each point on the dividing layer has the same surrounding as the neighbouring points which lie on the interface

Gibbs method of locating the dividing surface:chose surface such that surface density of atoms is zero in a one component system → Ns = 0 & N = N1 + N2

In a Multi-component system the surface density of the principal component is made zero by the choice of the surface

S

AB

CD

Interface

Phase-1 Phase-2

V1 V2

A

(AB & CD) S

dW → the reversible work done at constant (T, V, ) to increase the area by dA (without changing the volumes (V1 & V2) or states of each phase

, , , ,i i

S

T V T V

ddW

dA dA

( )d d F G is the change in thermodynamic potential which characterizes reversible work at constant (T,V,i)

S

S

Total SurfaceWork dA

The equilibrium shape of the interface will be given by the minimum value of the integral; such that no work is done on the bulk phases

Creation of interface under the constraint of constant chemical potentials implies the flow of species in and out of the control volume bound by ABCD

If dNi is added or removed from the interface:

ii

dNSurface Excess

dA

+ve or ve depending on if the species segregates or depletes at the interface

Pressure is not the same in two phases separated by a curved interface

An equilibrated system have two phases separated by a curved interface is characterized by T, V and (chemical potential)

Surface energy has a unique value only under equilibrium conditions

Surface Energy

Is the reversible work required to create a unit area of the surface at constant T, V, → increase in Helmholtz surface free energy.

A A AF E T S

A Liquid film has equilibrium surface configuration of atoms (or entities) specified by a certain concentration of atoms (surface density) with a surface energy

When a Liquid film is stretched, the surface will try to maintain this equilibrium configuration → atoms from the bulk will move to the surface to accommodate this increase in area (and maintain a constant surface density) → possible in liquid due to high atomic mobility

Additionally, the thickness of the film can adjust freely to avoid any volume strains in the liquid

Liquid vapor interface (or Liquid-liquid interface)

Work done in stretching the LIQUID film by dx

In terms of surface tension () In terms of surface energy ()

( ) 2

2

Work L dx

L dx

( ) 2

2x

x

Work L dx

L dx

dx

L

( )xF L

x

( )W Force displacement Area L dx

Increased separation compared to the bulk

Crude schematic!

The surface atoms show an increased separation as compared to the bulk

This is equivalent to a negative pressure (parallel to surface) → surface tension

The atomic displacements of surface atoms is such that stress to surface (z = 0) Liquid surface is in a Plane Stress Condition

Increase in surface area

Solids Liquids

Bond Stretching

Addition of surface atoms(from bulk)

Work required to increase area of a Liquid Create additional surface having same configuration

Work required to increase area of a Solid Create additional surface having same configuration Stretch bonds

SolidsSolids

Crystal = Bulk crystal + Surface crystal (with different atomic configuration than the bulk)

Taking the example of crystals

Surface crystal = Relaxed 2D crystal + Forces at the edges to match it with the bulk

+ + ForcesBulk Relaxed Surface Crystal

Surface viewed from top

Crystal(Solid)=

The forces can be tensile, compressive or shear (any general force) Force required is reduced by adjustment of atoms in 2nd and other layers below

the surface crystal ( some tangential forces have to be applied to the layers below to maintain equilibrium)

The real surface is a few layers deep!

The sum of all the forces (per unit length of edge) → gives the surface tension of the solid

If the surface structure is an extension of the bulk planar structure → no stresses are required for matching the 2D crystal to the remaining bulk Surface Energy Surface Stress

( )

x

y

xy yx

Solid surfaComponents of Surfac

ceincondition

of Plae Stress

ne Stress only

Effect of Symmetry of the Surface on the Stress Components

Across a line of Mirror Symmetry the shear stresses (xy) are zero

xy = 0

For a crystal surface with 3-fold or higher Rotational Symmetry the

normal stresses across all lines are equal and shear stresses (xy) are

zero

3-fold 4-fold 6-fold

Illustrated here for the case of 4-fold

3( )

-

2x yFor Surfaces with fold

Surface Stress SurfaceTensionSymmetry or higher

A cubic crystal having the same symmetry on the surface as in the bulk:4-fold {100} & 3-fold {111} surfaces

have no shear surface stresses and equal normal surface stresses

Relation between surface stress and surface energy in solids

Consider the following experiment

1+dx

1+dxdx

dx

½

½xyz

11

1

xyz

1+dx

dx

1

½

½

1

A

B

Stretch

Split

Stretch

Split

Assume:• Length in y direction is

constant during stretching• Centre of symmetry in the

crystal and the halves to be equivalent

A

B

Stretch Split

• Change in surface energy () on stretching → d new surface energy = ( + d)

• Work done on stretching → W0

• Strain on stretching (dx) = dx/1 = dx

Total Work Done Work Done to Stretch Work Done to Split

0 02( ) (1 ) 1 2( )(1 )xW W d dx W d d

StretchSplit

Total Work Done Work Done to Split Work Done to Stretch

12W W

• Work done on stretching the split haves→ W1

is different from the work done in stretching the unsplit haves due to surface stresses

[1]

[2]

• W1 W0 → is the work done by the surface stresses

[1] [2]From

0 12( )(1 ) 2xW W d d W

0 2( x xW W d d d d )[1]

1 0 2 2 ) 2x x xW W d d Work doneby the surface stresses d

xx

d

d

Similarly yy

d

d

12W W [2]

Due to this additional term x

B ShearSplit

A Shear Split

• For shearing process (area does not change during shearing)

Total Work Done Work Done to Shear Work Done to Split

0 02( )(1) 2 2W W d W d [1]

Total Work Done Work Done to Split Work Done to Shear

12W W [2]

1 0 2 xy xyWork doneby the surface stresses W W d

[1] [2] 0 12 2 2W W d W

1 0 2 2 xy xyW W d d xyxy

d

d

This term does not appear for liquids

xyxy

d

d

xx

d

d

Surface Energy Surface Stress only if does not change with the stretching process

The equality of and depends on the ability of the surface to maintain its configuration while stretching → i.e. on the mobility of the atoms and the relaxation time required for the surface atoms to regain their undistorted configuration by atomic migration

Liquids: trelaxation << tstretch

In crystals for some disordered boundaries: trelaxation ~ tstretch → the boundaries behave as liquid films

A crystal plane at a angle to a close packed plane will have will have additional bonds broken as compared to the close packed plane

Such a surface can be described in terms of ledges and terraces A general surface described interms of two orientations ( & ) will consist of

ledges and kinks (in the ledges) Any general orientation within the stereographic triangle (Euler triangle) can be

constructed with a ledges and kinks of certain density in an appropriate terrace orientation

Anisotropy in Surface Energy

ACos

S A

SCos

LATan

A LATan A

( ) T T L LEnergy A A

Area S

( ) T LA ATan

ACos

( ) T L T LTan Cos Cos Sin

( ) T LCos Sin

Note: the origin of is due to “broken bonds”!

Equation of circle passing through origin 2 ( )r RCos

2 ( )

(2 ) (2 )

r R Cos Cos Sin Sin

RCos Cos R Sin Sin

( ) T L T LTan Cos Cos Sin Comparing with:

(2 )T RCos 2 2 2 2 2 2 2(4 ) (4 ) 4T L R Cos R Sin R

2 2 2T L R The diameter of the circle:

(2 ) (2 )r RCos Cos R Sin Sin

( ) is the equation of a circle passing through the originT LCos Sin

2 2With diameter (2R) = T L

(2 )L R Sin L

T

Tan

2 2 112Centre of Circle O , L

T LT

Tan

( ) T LCos Sin

-plot or Wulff plot

Entropy effects are ignored so far → when included the cusps could be less prominent and could even disappear for high index planes

In this simplistic model the energy of the ledge L is assumed independent of the ledge spacing

In reality some ledge interaction will be present L will be a function of ledge spacing (and thus of the surface orientation)

ledges will be observed for all rational orientations

The orientation dependence of will tend to rotate the surface to a low energy orientation → produce a torque on the surface

( )Torque term

( )

From -plot to EQUILIBRIUM SHAPE OF CRYSTAL → the Wulff construction

Draw radius vectors from the origin to intersect the Wulff plot (OA in Figure) Draw lines to OA at A (line XY) The figure formed by the inner envelope of all the perpendiculars is the

equilibrium shape

Wulff plot → Equilibrium shape From the equilibrium shape → it is not uniquely possible to construct a Wulff

plot Wulff plot with sharp cusps equilibrium shape = polyhedron Width of the crystal facets 1/(surface energy)

→ largest facets are the ones with lowest energy

FCC

The picture below shows a water droplet on a plant leaf. Note that the droplet has beaded up. A schematic of the picture is shown in the diagram, with surface (interface) tension forces included. There are 3 interfaces and correspondingly 3 forces.

The angle that the tangent to the droplet lens at the triple line is called the contact angle and this angle can be calculated using force balance as below (eq. (1)).

Contact Angle

Cos

is the contact angle

(2)

CosSurface tension force balance (1)

The contact angle changes depending on the substrate (keeping the liquid constant- water for now).

For most leaves the upper side (adaxial) is less hydrophobic (with a lower contact angle) as compared to the lower side (abaxial) which is more hydrophobic (with a higher contact angle). In lotus leaf the upper side is more hydrophobic.

Water on lower side of banana leaf

Water on upper side of banana leaf

Water on glass slide

Water on guava leaf

Water on lotus leaf

Water on lower side of pipal leaf

A closer look at the upper side of the lotus leaf!