Surface Energy, Surface Tension & Shape of Crystals

Transcription

1 Surface Energy, Surface Tension & Shape of Crystals

2 Shape of Crystals 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. Dendritic growth of crystals Close to equilibrium shape Note the facets Electrodeposited nanocrystalline Al-Mg alloy powders (Photo courtesy: Dr. Sankarasarma Tatiparti) Video: Video: Dendritic Dendritic growth growth of of crystal crystal from from melt melt KDP crystals grown from solution

3 What is a surface and what is an interface? 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). Creation of a surface Cut and Separate (or materials) Creation of an interface Join (or materials) Note: Surface can also be thought of a vacuum-material interface (or even a air-material interface)

4 How to understand surface energy? We use a crystal to understand the concept 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?!! 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 E b per bond. Each bulk atom is bonded to 4 atoms (as in the upcoming figure) Energy lowering of bulk atoms = 4E b this is negative energy w.r.t to the unbonded state Each surface atom is bonded to 3 other atoms only Energy lowering of bulk atoms = 3E b this is also negative energy w.r.t to unbonded state! Cotd...

5 Energy lowering on the formation of infinite crystal/unit volume = [ (number of atoms) 4E b ] Energy of a crystal with a free surface/unit volume = [ (number of atoms) 4E b ] + [(number of surface atoms) 1E b ] An alternate calculation without invoking surface energy This is the surface energy! Energy of a crystal with a free surface/unit volume = [ (number of bulk atoms) 4E b ] [(number of surface atoms) 3E b ] The reference state for the surface energy is the bonded state and not the free state Schematic not to scale 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!

6 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).

7 Surface Energy 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.

8 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

9 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 () Let us look at the units of these two quantities Force F [ N] Energy E [ J] [ Nm] [ N] 2 2 Length L [ m] 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 LIQUIDS Surface Energy Surface Tension SOLIDS Surface Energy Surface Tension Except in certain circumstances + Surface energy is Anisotropic

10 A comparison of the solid and liquid surfaces LIQUID SURFACE Surface Energy Surface Tension Characterized by one number the surface density Liquids cannot support shear stresses (hence use of the term surface tension) Surface Energy Surface Stress (Tensor) Surface Torque SOLID SURFACE Has a structure and hence more numbers may be needed to characterize a solid surface Crystalline surfaces all the lattice constants will be required Amorphous surfaces Density + a Short Range Order parameter In the case of solids the term surface tension (which actually should be avoided) refers to surface stresses

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

12 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

13 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.

14 Some more mathematical looking concepts! Some readers may want to skip the pages with too much math and get to pages of interest.

15 Surface/Interface Effects become important (Surface : Volume) is large Interface has large curvature 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

16 1 2 Variation of thermodynamic function across the interface 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 dc F An v [ f ( c) k ] dx dx Gradient term: Contribution due to variation in composition 2 F Helmholtz free energy f(c) F per molecule of a homogenous system of composition c n v No. of atoms per unit volume A Cross sectional area k Constant for small gradients

17 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 N s = 0 & N = N 1 + N 2 In a Multi-component system the surface density of the principal component is made zero by the choice of the surface

18 dw the reversible work done at constant (T, V, ) to increase the area by da (without changing the volumes (V 1 & V 2 ) or states of each phase dw ds da da dd( F G) TV,, TV,, Total SurfaceWork i i is the change in thermodynamic potential which characterizes reversible work at constant (T,V, i ) S S da Interface A D S Phase-1 A V 1 V 2 B C Phase-2 (AB & CD) S 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

19 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 dn i is added or removed from the interface: Surface Excess i dni da +ve or ve depending on if the species segregates or depletes at the interface

20 Surface Energy 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 Is the reversible work required to create a unit area of the surface at constant T, V, increase in Helmholtz surface free energy. F E T S A A A

21 Liquid vapor interface (or Liquid-liquid interface) 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

22 Work done in stretching the LIQUID film by dx In terms of surface tension () In terms of surface energy () Work ( L dx) 2 x 2 x Ldx Work ( L dx) 2 2 Ldx F ( x L) W ( Force displacement) Area L dx x L dx

23 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 Increased separation compared to the bulk Crude schematic!

24 Increase in surface area Solids Addition of surface atoms (from bulk) Liquids Bond Stretching 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

25 Solids Taking the example of crystals Crystal = Bulk crystal + Surface crystal (with different atomic configuration than the bulk)

26 Surface crystal = Relaxed 2D crystal + Forces at the edges to match it with the bulk Surface viewed from top Bulk + Relaxed Surface + Forces Crystal = Crystal (Solid) The forces can be tensile, compressive or shear (any general force) Force required is reduced by adjustment of atoms in 2 nd 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

27 Components of SurfaceStress xy x y yx Solid surfaceincondition of Plane Stress ( only)

28 Effect of Symmetry of the Surface on the Stress Components Across a line of Mirror Symmetry the shear stresses ( xy ) are zero xy = 0

29 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 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 For Surfaces with3-fold Surface Stress SurfaceTension ( ) Symmetry or higher x 2 y

30 Relation between surface stress and surface energy in solids Consider the following experiment 1+dx Stretch Split 1+dx A z y x dx ½ 1+dx dx z y x B ½ 1 Stretch ½ dx Split ½ 1 Assume: Length in y direction is constant during stretching Centre of symmetry in the crystal and the halves to be equivalent

31 A Stretch Split Change in surface energy () on stretching d new surface energy = ( + d) Work done on stretching W 0 Strain on stretching (d x ) = dx/1 = dx Total Work Done Work Done to Stretch Work Done to Split W W 2( d) (1 dx) 1 W 2( d)(1 d ) 0 0 x [1] B Split Stretch Total Work Done Work Done to Split Work Done to Stretch [2] W 2 W 1 Work done on stretching the split haves W 1 is different from the work done in stretching the unsplit haves due to surface stresses

32 W 1 W 0 is the work done by the surface stresses [1] W W d d dd ) 0 2( x x [2] W 2 W 1 From [1] [2] W W 2( d)(1 d ) 2 W 0 x 1 W1W 0 2d 2 dx ) Work doneby the surface stresses 2x dx x d d Similarly y x d d y Due to this additional term x

33 For shearing process (area does not change during shearing) A Shear Split Total Work Done Work Done to Shear Work Done to Split W W 2( d)(1) W 2 2d [1] 0 0 B Split Shear Total Work Done Work Done to Split Work Done to Shear W 2 W [2] Work doneby the surface stresses W W d xy xy [1] [2] W W 0 2 2d 2 W 1 W 1 W xy 0 2d 2 xy d xy d d xy This term does not appear for liquids

34 x d d x xy d d xy 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: t relaxation << t stretch In crystals for some disordered boundaries: t relaxation ~ t stretch the boundaries behave as liquid films

35 Anisotropy in Surface Energy 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

36 Cos A S S A Cos Tan AL A ATan A L Energy T AT L AL ( ) Area S ( ) A ATan T L A Cos Tan Cos Cos Sin ( ) T L T L ( ) T Cos L Sin Note: the origin of is due to broken bonds!

37 Equation of circle passing through origin r 2 RCos( ) r 2 R( Cos Cos Sin Sin) (2 RCos) Cos (2 RSin) Sin r (2 RCos ) Cos (2 RSin) Sin Comparing with: T L Tan Cos Cos Sin ( ) T L T L (2 RCos) L Tan T L (4 R Cos ) (4 R Sin ) 4R (2 RSin) T The diameter of the circle: 2 2 T L 2R ( ) Cos Sin is the equation of a circle passing through the origin T 2 2 With diameter (2R) = T L L L Centre of Circle O 2 T L, Tan T

38 -plot or Wulff plot ( ) T Cos L Sin Entropy effects are ignored so far when included the cusps could be less prominent and could even disappear for high index planes

39 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 ( ) ( )

40 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

41 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

42 FCC

43 Contact Angle 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)). Cos Surface tension force balance (1) Cos (2) is the contact angle

44 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 glass slide Water on upper side of banana leaf Water on lotus leaf Water on guava leaf Water on lower side of pipal leaf

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

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