1. 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 of the surface of area A, the area change dA is a product of that portion of perimeter and the length moved. Work term - dA; force x distance, and appear in the combined 1 st and 2 nd laws of thermodynamics as J. W. Gibbs, collected works, Yale Univ. Press, New Haven, vol.1(1957), p. 219 ~ 331.

2. For a system containing a plane surface this equation can be reading integrated : Strictly speaking, is defined as the change in internal energy when the area is reversibly increased at constant S, V and N i (i.e., closed system). where U – TS + PV is the Gibbs free energy of the system. and rearranging for yields.

3. And is the Gibbs free energy of the materials comprising the system. def Surface Excess Quantities Macroscopic extensive properties of an interface separating bulk phases are defined as a surface excess. Thus is an excess free energy due to the presence of the surface.

4. There is a hypothetical 2D “dividing surface” defined for which the parameters 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 present in the same volume if the phases were homogeneous right up to the “ Dividing Surface ” i.e., The real value of x in the system The values of x in the homogeneous and phases

5. 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 = Then letting A = a 2 where a lattice spacing 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.

6. pair potential r U(r) and If the solid is sketched such that the surface area is altered the energy The total energy of the surface is changed by an amount.

7. Surface Stress and Surface Energy Then in general the relationship between surface stress and surface energy is given by, For a surface with 3-fold or higher rotational symmetry f ij is isotropic and the surface stress can be treated as a scalar.

8. 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. for an isotropic surface For an anisotropic surface f is a tensor quantity and where double sum in the strains and.

9. The anisotropy of surface energy Surface energy is a function of orientation – crystalline solid. For a liquid γ is isotropic and the equilibrium shape minimizes the surface / volume ratio. For example, the equilibrium hape of a soap bubble is a sphere. Experimentally it has been found that cuts of crystals off a low index orientation equilibrate to form stepped structures such that the steps are composed of low-index surfaces. broken bond ( 1 0 ) plane

10. Bond density along direction defined by θ is greater than the bond density along a low index direction owing to step structure. 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:

11. (10) “cusps” @ Polar Plot of Low-index plane have cusps in plots at 0 K which tend to get rounded off at higher temperatures.

12. Inner envelope of normals defining the equilibrium of shape of crystal. Equilibrium shape of a crystal obtained when and this is given by the Wolff’s theorem. ( 0 1 ) ( 1 1 ) ( 1 0 ) Example of a 2D polar plot of

13. 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.

14. 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 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.

15. Faster growth Slower growth Faster growth

16. 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.

17. Estimation of interfacial energies Recall: 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.

18. For the liquid / vapor interface - For the solid / liquid interface – entropic effects dominate. Note the temperature dependence. For many situations, these values provide reasonable estimates. For the solid / vapor interface -

19. ( J / m 2 ) near T m ( J / m 2 ) @ 25ºC Sn0.68MgO1.0 Ag1.12 CaF 2 (111) 0.45 Pt2.28 CaCo 3 (1010) 0.23 Cu1.72 LiF (100) 0.34 Au1.39 NaCl (100) 0.30 T (°C) ( J / m 2 ) H2OH2O250.072 Pb3500.442 Cu11201.270 Ag10000.920 Pt17701.865 NaPO6200.209 FeO14200.585 Al 2 O 3 20800.700 Note that near Tm,  lv ~  sv. Values of

20. ( J / m 2 ) near T m Al0.093 Cu0.177 Fe0.204 Pb0.033 C 2 H 2 (CN) 2 [succinonitrile]0.009 Nylon0.020 and for metals Values of (inferred from nucleation exp) Temperature dependence of ( solid / vapor, liquid / vapor ) recall that for a 1 component system:

21. Solid / Solid Interfaces (i)Chemical bonding (ii) Structural bonding ( say phases have different crystal structure) y (distance) xαxα xβxβ x α is the mole fraction of A in α x β is the mole fraction of A in β Chemical contribution to : Consider a general inter-phase α/ β boundary. can be thought of as being composed of 2 terms :

22. Let U(x 1, x 2 ) be the sum of the bond energy per unit area between planes of composition x 1 and x 2 where V AA, V BB, and V AB are the bond energy. In analogy with the regular solution model we define an excess energy, U i, due to the interface: where Ω can be estimated from Ω/2R = T critical (see Reg. Sol. Theory) Regular solution model of an interfaces (Becker 1938)

23. For metal / metal interfaces : For close to pure metal interfaces : so Typical values of lattice matched (coherent) interfaces energies range from 10 -3 ~ 10 -1 J/m 2.

24. * The diffuse chemical interface (1-D estimates) – variation in the mole fraction of A atom 1 plane to the next  interplaner spacing y (distance)

25.  generalize to a continuum  dx a  dy composition gradient K 1  Gradient energy coefficient in 1D * important in spinodal decomposition

26. 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 of two terms: (i) g 0 (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 compositional gradient  K(  C) 2 The total free energy is expressed as a functional ( a function of a function) i.e., for metals K  10 -19 J/m

27. 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)

28. Defmisfit 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.

29.  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

30. Cut ABCD Consider the triple point of a gb junction: Grain 1 Grain 2 Grain 3  12  23  13  12  23  13 Rot @ y-axis Twist boundary Tilt boundary Rot @ x-axis A y z x B C D

31. Herring (1951) showed that by balancing the forces for a virtual change in the orientation of the triple junction : where theterms are called “ surface torque” terms. For high angle gb the torque terms can be neglected and d  23 d  13

32. Wetting (Contact) Angle  LV solid  SV  SL  Force Balance Def  is called the wetting or contact angle.

33. θ = 0 Complete wetting θ < 90º θ > 90º θ > 180º No wetting