9 Chapter 9 General Properties of Crystalline Solid Surfaces ■ Types of Surfaces of crystalline solids： Free Solid Surface：

Chapter 9

General Properties of Crystalline Solid Surfaces 

■ Types of Surfaces of crystalline solids：

Free Solid Surface：external surface exposed to the environment ( vapor, gas, vacuum )

Internal Surface ( grain boundary )：the boundary between two orientations of the same material

■ The components involved to express the surface free energy (dG^σ) of crystalline solids： 
 

    The 1^st term：γdA→ surface tension effect

    The 2^nd term：surface chemistry, includes

      ：chemical potential, depends on surface composition

        ：effect of shear stress (σ), strain (ε) 

■ Major differences exist between fluids and solids：

   1. solids resist shear stress and can be elastically strained.

2. the solid crystalline state is characterized by long-range order and consists of a regular structure array of atoms.

     → anisotropic behavior, variation of properties with orientation.

3. the kinetics are slow and the solid-solid interface can “freeze” into a metastable state. 


1. Crystalline Solids Consists of a Regular Array of Atoms

9.1.1－Each crystalline material is defined by its unit cell and lattice parameters. 


• Lattice points can be arranged to fill space in only fourteen different ways (known as Bravais lattices), which can be further organized into seven crystal systems ( Fig.9.1).
• Unit Cell：the smallest repeat unit of the space lattice. 
  

           lattice parameters：a , b , c

           lattice angles：α , β , γ

perfect crystalline：the characteristic length over which order is maintained is infinite. 
 
 


Figure 9.1 Conventional unit cells of the 14 Bravais space lattices 


• three types of interface can be distinguished by features that change across boundary
1. Interphase boundary：crystalline phase changes completely across boundary － unit cell, lattice parameter, and orientation all change.
2. Epitaxial boundary：lattice parameter, unit cell change, but orientation relationship exists.
3. Grain boundary：only orientation changes.
 


9.1.2－Close-Packed Arrays of Spherical Molecules Bonded by Dispersion Forces ( Inert Gas Solids )

• For inert gas solids, the free energy in the condensed state is minimized when each atom is surrounded by the greatest possible number of nearest neighbors ( coordination number )
• When all atoms are the same size, the highest coordination number is 12, and the relevant structures are：

           hexagonal close-packed (hcp)

           face-centered cubic ( fcc ) or cubic close packing ( ccp )

• A close packed layer of equal-sized spheres
• Atoms in the second layer may occupy either P or R positions, but not both together 
  
• Addition of a third close packed layer can be done in two possible ways, S or T positions in Fig. 7.2 , and thereby lies the distinction between hcp and ccp. 
  

hcp：the third layer is packed at S → ABABA…

ccp：the third layer is packed at T → ABCABC... 


• The 12 coordination number in a close packed structure：

            6：coplanar with the center sphere

            3：in the plane above

            3：in the plane below  


• The unit cell of ccp and hcp
• ccp → face centered cubic ( contains 4 atoms )
• hcp→ hexagonal unit ( contains 2 atoms ) 
  
 


9.1.3－Ionic Solids, Basic to Many Ceramics, Are Composed of Different Sized Positive and Negative Ions Bounded by Coulombic Forces

• Cations (+ze) and anions (-ze) are held together on crystal lattices by coulombic forces.

----- nondirectional interaction：

----- free energy is minimized when each ion is in contact with the highest number of oppositively charged ions. 


• Because the cations and anions are not the same size, and may not have the same valence, the relative coordination number of anions and cations may not be equal.

----- arrange into configurations that provide the highest possible coordination consistent with the relative size and valence of the ions.

• cations are surrounded by the maximum number of anions
• anions are surrounded by the maximum number of cations
• coordination number depends on the cation-anion radius ratio 
  

Figure 9.3 Effect of ionic radius ratio on coordination number 


• The relative number of ions in the crystal structure must equal the stoichiometric composition of the ionic compound.-----Pauling’ rules

      


Figure 9.4 Crystal structures with six-fold coordination (a) The sodium chloride crystal structure. (b) Depiction if sodium chloride crystal structure with ions drawn to scale

• To determine the total interaction potential------

evaluate all the interaction of one ion ( ex. Na⁺ ) with all others in the crystal ( ex. Na⁺ and Cl^－ ).

• Example：NaCl, for each Na⁺ ion

6     nearest   Cl^－ at R=0.276 nm

12  2^nd nearest  Na⁺ at d=2^1/2R

8   3^rd nearest  Cl^－ at d=3^1/2R

6   4^th nearest  Na⁺  at d=4^1/2R

    The bonding potential for a given ion：

                  ( 9.1.1 )

          1^st term：attractive； 2^nd term：repulsive

          opposite sign → V = ”－”

              same sign → V = “＋” 


• the summation can be represented by Madelung constant (α), Eq 9.1.1 becomes

                    ( 9.1.2 )

         R：the minimum separation of opposite ions

for NaCl： 


Since |z₁| = |z₂| = 1 


• Lattice energy per mole is V_bond N_AV = 880 KJ / mole ,

This energy is 15% higher than the measured values, due to the neglect of the repulsive term (C / R¹²).

• The Madelung constant (α)：

       ----- always greater than 1 ,

       ----- the value is dependent on the crystal structure

       (for 1：1 stoichiometry ionic solid)

       α = 1.763 ( highest ) for CsCl ( coordination no. = 8 )

       α = 1.748 for NaCl ( coordination no. = 6 )

       α = 1.638 for ZnS ( coordination no. = 4 )

         ----- for 2：1 salt like CaF2 , α increases to 5 


• Ionic solids generally have high melting points and high latent heats of melting due to the high strength of the ionic bond.
 


9.1.4－Covalent Bonding Plays a Key Role in Determining the Structure of Many Materials Used in Electronic Devices, Such as Diamond and III-V and II-VI Compounds 


• Electrons from two or more atoms are shared together so that each atom has the desired outer－shell configuration
• Covalent bonds are strongly directional, that is, they are exerted in specific angular directions.
• For diamond structure of carbon atoms：

Outer orbitals (2s², 2p²), the four electrons become hybridized to form four asymmetrical orbitals ( sp³ ) ----- ( Fig. 9.5a ) 


Figure 9.5 Crystal structures with four-fold coordination. (a) The diamond crystal structure for covalently bonded carbon. (b) The zinc blende (ZnS) crystal structure. (c) The zinc blende crystal structure of (b) observed at right angles to the [111] axis and along a [110] direction. This view highlights the alternate Ga and As layer structure in gallium arsenide and the asymmetry in layer spacing. 
 


• Other materials in which the covalent bond plays a prominent role are：IV-IV ( SiC, share 4+4 electrons )；

        III-V ( GaAs, share 3+5 electrons )；

        II-VI ( CdS, share 2+6 electrons )

Ex：ZnS ( Fig. 9.5b ) ----- the same crystal system are diamond except that the atoms located at the central and the corners of the tetrahedral configuration are unlike. 


9.1.5－The Special Properties of Metals Arise from the Metallic Bond in Which Valence Electron Are Shared by All Atoms 


• Metallic crystals are pictured as an array of positive ions, embedded in a sea of mobile electrons. Metals display feature of both covalent and ionic bonding

Covalent feature：metal ions share electrons with total community of other ions

Ionic feature：the interaction between metal ions and electrons is Coulombic attraction

• The electrons are no longer localized → nondirectional
• All the metallic ions have the same size in pure metals, they prefer to form close – packing structure, with 12-fold coordination 
  

Ex：hcp array：Zn    

    Fcc array：Cu, Al, Au

• For alloys, the structure are similar to those of ionic solids
 
 


9.1.6－Molecular Solid Exhibit Mixed Bonding

• Molecular units (SO₄^-2, CO₃^-2 ) bond to other molecular units by ionic, covalent, or van der Waals force.
• Ionically bonded：SO₄^2-( covalently bounded molecular units ) to Mg⁺²；or Mg₂SiO₄
• Polar bounded：for molecular units do not carry residual charge. ( Ex：H₂O molecules from ice crystals )
• Mixed bounding：polar bounding ( CdCl₂, MoS₂ ) or covalent bounding ( graphite ) between molecules in the plane, and van der Waals bounding across the planes ( Figure 9.6a：graphite ). Such “layered” structure can be used as solid lubricants.
• Mica, K(Al₃Si₃O₁₀)(OH)₂：multiple bounding

----- (OH^－)( covalently bounded molecules ) are attracted to Al, Si, and O by ionic or covalent attraction to form a layered structure.

----- the layers are held together by electrostatic attraction to potassium ions ( Figure 9.6 b)  
 
 
 
 
 
 
 
 


9.2 Characteristics of the Free Solid Surface

9.2.1－The Free Surface Energy Equals One-Half the Cohesive Energy

                                   ( 9.2.1 )

γ_SV：surface energy at the solid-vapor interface

• the difference in surface characteristics between solid and liquid.
 


    ----- For solid：

1. γ_SV is anisotropic, depends on orientation
2. surface energy and surface tension are not necessarily equal for a solid.
3. The solid surface may not be atomically flat.
 


9.2.1.1－Surface Energy Depends on Bond Character and Crystal Structure and Is Generally Higher for Crystalline Solids Than for Amorphous Solids and Liquids.

• The excess free energy per unit area of surface, ( dG^σ/dA ) can be estimated from the “dangling” bond of the atom located at the interface.
• Example：NaCl crystal. When the crystal is broken in two through ( 100 ) surface, each atom losses one at its six nearest neighbors.→ one dangling bond per surface atom.
 


                or                 (9.2.2) 


• For metal in closed packed configuration ( hcp , fcc )：       
• for exposing the closed-packed planes ( {0001} for hcp, and {111} plane for fcc ), three of the nearest neighbors should be split to leave 3 dangling bonds per atom. On sublimation, the remaining 9 bounds are broken. So,
 


       or                    (9.2.3)

• For metal with interatomic spacing ～3Е,

    U_sub.＝335 KJ╱mole , γ_SV～1-2 J╱m² or

    1000~2000 erg╱cm² (dyne╱cm)

• Compare Eqs (9.2.2) and (9.2.3)

    γ ( metal ) ＞ γ ( ionic material ) 


• The broken bond approximation for estimation surface energy is valid for bounding of short range interaction (metallic and van der Waals bond ). For ionic solids, the Coulombic attraction is long ranged and the surface energy must be calculated by summing all the attractive interaction and comparing the result with the bulk lattice energy. Coulombic attraction contributes nearly all the bulk lattice energy ( sublimation ), van der Waals attractions contribute～30% of the surface energy.
• When the surface energy is measured by the “fracture mechanism” technique, it is termed fracture surface energy, γ_f. Because some of the strain energy is consumed in other ways ( such as plastic deformation ),
 


For brittle solid ( like ionic solid)：γ_f differs only a little to γ_SV

For less brittle solid：γ_f may rise to a value many orders of magnitude above γ_SV

• Surface energy of a solid depends on：

    Crystal structure ( coordination number )

    Bond character ( strength )

    Orientation of surface ( density of broken bonds )

----- metal：high coordination number, high strength 


-----ionic material：γ increases with increase of valence 
 


-----

• A solid with a high surface energy would prefer to be covered by a solid with a lower surface energy.

----- ceramic or polymer on metal → easy

    Metal on polymer → difficult

9.2.1.2－The Surface Energy of a Crystalline Solid Depends on Orientation and Is Anisotropic

• Surface energy is lowest for the closest-packed planes because the number of bonds lying within the plane is the highest, and the number of bonds across the plane is at the lowest. → the broken bonds left dangling is at a minimum.

Ex：{111} planes of fcc metal

• Equilibrium shape：

           liquid → minimized surface area ( spherical droplet )

solid → terminated in surfaces with the lowest energy predicted by Wulff Plot.

• Wulff Plot presents the variation of surface energy with orientation ( Fig 9.7 ). A particular plane is represented by a vector drawn from the center point of a unit cell in a direction normal to that plane
• The length of the vector is proportional to the surface energy of the planar surface.
 
 
 
 


Figure 9.7 Surface energy is anisotropic for crystalline solids. (a) Wulff plot of surface energy for different directions in the NaCl structure, showing construction of the surface with minimum surface energy, in this instance a cube. (b) Wulff plot of surface energy for different directions in the {110} plane of the face-centered cubic crystal structure, showing construction of the surface with minimum surface energy. (c) Three-dimensional surface of minimum surface energy for face-centered cubic crystal structure in (b), in this instance a truncated octahedron ( almost a sphere ).. 
 


• The symmetry of the crystal structure affects the orientation sensitivity of the surface energy.

        Ex：GaAs - noncentrosymmetry (Fig 9.5C) 


Figure 9.5 (c) The zinc blende crystal structure of (b) observed at right angles to the [111] axis and along a [110] direction. This view highlights the alternate Ga and As layer structure in gallium arsenide and the asymmetry in layer spacing. 


• Cleavage between {111} plane produce two different kinds of surfaces:
1. Ga terminated: 3 arsenic(As) neighboring beneath to share its valence electrons (3 electrons), free from dangling bonds → {111}_Ga has low surface free energy.
2. As terminated: has 5 valence electrons, and only 3 Ga neighbors at the surface, two bonds are left dangling. → {111}_As has high surface energy. 
   
• Acid etch the {111}_As surface more easily than {111}_Ga，As surface is more favorable for rapid vapor deposition of new material than Ga surface. 
  

 9.2.1.3－Surface Energy and Surface Stress Are Not Equivalent in Solids 


• Surface energy：excess free energy needed to form an unit area.

Surface tension：force per unit length opposing the formation of new surface.

    For liquid：surface energy = surface tension

For solid：the equivalence no longer true because a solid can sustain a state of stress.

• For liquid, surface energy is used to bring atoms from the bulk to form the new surface. In a solid, strain energy is spend strecting the old surface without bring new material to the surface.→ Surface stress (surface tension) exceeds the surface energy.
 


9.2.2－The Free Solid Surface Has a Structure Other Than That Rendered by Simple Dissection of the Material into Two Halves 


9.2.2.1－Surface Atoms Rearrange Themselves to Minimized Free Energy–Surface Relaxation and Surface Reconstruction at Low Temperature

• The process of rearrangement is controlled by：

    Bonding, crystal structure , magnitude of the driving force. 
 


• The ability of atomic arrangement depends on the size and sign of ions involved.

       Ex：Zinc oxide (ZnO), an ionic crystal.

          Oxide ions are 1.5 times the diameter of Zinc ions. Zinc ions may retreat to semi-interstitial location just beneath the surface where they are shield from complete surface exposure. (Fig 9.8) 
 


• Rearrangement is more pronounced in covalent crystals because the bonds are so strong and directional.

    Ex：Si(111) surface → ( 7×7 ) surface reconstruction

        Si(100) surface → ( 2×1 ) (Fig 9.9) 
 
 
 


Figure 9.9 Surface reconstruction on the surface of silicon；(a) planar view and (b) cross section of (100) surface. Left-hand portion shows atomic configuration before reconstructed and right-hand after reconstruction. Surface energy is minimized by small movements of Si ions toward each other along [100]. 


• The reconstructed state is an important consideration in substrate behavior when new layers are to be added to a surface.
 


9.2.2.2 Surface Roughening at High Temperatures 


• A clean surface develops a texture or roughness on an atomic scale. The surface structures can be classified as：terraces, steps , kinks , adatoms , vacancies. (Fig 9.10)
• The energy associated with each surface site depends on the coordination number. 
   
   
   
   
   
  

        Coordination number decreases → Energy increases.

• Atoms in the higher-excess-energy ( low coordination number ) locations tend to move to location of low energy by：(1) surface diffusion , (2) vaporize to the vapor. Thus, a solid surface is atomically rough and continuously changing at high temperature.
 


9.3 The High Surface Energy of a Crystalline Solid Is Reduced through a Change in the Chemical Composition of the Surface 


• At low temperature (T < 1/2 T_m, the meeting temp. )：

    surface chemistry is influenced primarily by adsorption.

• At high temperature (T >> 1/2 T_m)：

influenced by the diffusion of impurities from inside the crystalline solid to the surface. --- (segregation). 


9.3.1 Adsorption Reduces Surface Energy at Low Temperatures 


• Even in the best vacuum systems, remnant gas ( O₂,N₂,HC,…) are unavoidably present and adsorb on a solid surface.
 


9.3.1.1 The Langmuir Adsorption Isotherm Describes Adsorption of the First Monolayer 
 


      Rate of adsorption: k_AP (N - n), Rate of desorption: k_Dn

         k_AP (N - n) = k_Dn   

          K_eq = k_A/k_D = n/P(N-n) = θ/P(1-θ)

       Where  θ=n/N

Langmuir equation：

                     (9.3.1)

Θ：coverage ratio of the available sites.

A：pressure of gas (P) or molar concentration of solute (C)

K_ads：equilibrium constant

• For gas：(Fig 9.11a)

                                    (9.3.2)

                   (9.3.3)

α：accommodation coefficient–a measurement of the interaction between the gas atom and the surface atom.

1/τ₀：vibration frequency of solid atoms.

M：molecular weight.

ΔG_ads：energy of adsorption. 
 
 


• Clausius–Clapeyron eq.
 


             (9.3.3)

       (Fig 9.11c) 
 


9.3.1.2 A Distinction between Physisorption and Chemisorption Can be Based on Adsorption Energy

• Physisorption：weak physical attraction of the gas ( the adsorbate ) to the crystalline solid surface ( the adsorbent)

    ----- ΔG_ads < 60KJ/mole (0.6 eV / atom)

    ----- reversible process, can be recovered by heating.

        ex：reactive a charcoal filter by heating. 


• Chemisorption：Strong Chemical reaction involved irreversible.
• 1. Low ΔG_ads ( ≦ 6 kJ/mole, 0.06 eV / atom)：

         Inert gas (Ar, He) on ionic solid surface (KCl).

       2. Moderate ΔG_ads ( 30 kJ/mole , 0.3 eV / atom )：

         Polar molecular (NH₃) on inert charcoal surface.

         ----- Strong physisorption. 


    3. High ΔG_ads ( 65 kJ/mole , 0.65 eV/atom)：

      Active carbon monoxide (CO) on Cu surface.

      ----- Chemisorption.

    4. ΔG_ads > 100 kJ/mole (1 eV/atom)：

      Associated with active surface oxidation of metals -----         a chemical reaction process.

• At normal ambient temperature and pressure, physisorption results in a complete monolayer almost instantly. Even at low vacuum pressure (10^-6 torr), a monolayer forms within a millisecond.
• Chemisorption may not be instantaneous. It often takes place in two stages：(1) the physisorption of the adsorbate, (2) a chemical reaction to form a compound.
 


9.3.1.3 The BET Equation Describes Multilayer Coverage of Solid Surfaces.

• The molecules of the first monolayer were strongly attracted to the surface, subsequent molecules may experience a weak attraction, or even a repulsion.
• ΔG_ads,l ( the energy of adsorption for the first layer ), will be greater than the energy of adsorption for all succeeding layers (ΔG_ads,v).

   (ΔG_ads,v ～ heat of condensation of the adsorbate.) 
 


• The BET equation, ( by Brunauer, Emmett, and Teller )：

               (9.3.5) 


              ( partial pressure / equilibrium pressure ) 


9.3.2 Segregation Reduced Surface Energy at High Temperature

• At high temperature, the kinetics of surface adsorption change so that the surface chemistry is influenced as much by the internal environment as by the external one.
• Inside the solid, rapid diffusion of impurities enables the solid to approach thermodynamic equilibrium. The reduction in surface energy (due to the presence of solute) affects the surface chemistry.
• Timescales for the diffusion of impurities to come to equilibrium.

         Solid → seconds.

         Liquid → microsecond.

9.4 Crystalline Solids Contain Many Kinds of Imperfections

1. Point imperfections：vacancies and interstitials (zero – dimension imperfections )

2. Linear imperfection：dislocation (one – dimension imperfection )

3. Planar imperfection：small – angle boundaries；stacking faults ( two – dimension imperfection )

4. Volume imperfection：voids ( 3 – D imperfection.)

9.4.1 Point Imperfection Exist in Thermodynamic Equilibrium 


• Perfect crystal can exist only at absolute zero temperature. The ΔG of a solid is reduced when point defects (vacancies and interstitials) are introduced. ( Fig 9.12)
• Total free energy of solid ( G )： 
  

                                    (9.4.1)

G^o：standard state free energy in the absence of imperfection

     n：number of imperfection

     ΔH_f：enthalpy required to form imperfection

     ΔS：entropy due to the presence of imperfections

 


     

                       (9.4.2)

     N：number of lattice sites , n：number of imperfection

     Substituting Eq.(9.4.2) into Eq.(9.4.1) 


     assume ,

     at equilibrium

• The equilibrium concentration of point imperfection is：

                                      (9.4.3) 


• For metals , ΔH_f ～1eV

9.4.1.1 Free Surfaces Act as Both Sources and Sinks for Point Imperfections

• ΔH_f of metal is about 1eV, equivalent to

               40 kT at room temperature

               13 kT at 600℃ 

        Far above the thermal energy (～kT)

• As temperature rises → additional vacancies must be created.

        As temperature falls → vacancies must be dissipated. 


• The easiest place for this to occur is at the free surface. The surface thus acts as the source for additional vacancies as the temperature rise, as sink for reducing the concentration of point imperfections as the temperature falls.
• The time needed to reach equilibrium of point imperfection is determined by Einstein relationship：

                                         (9.4.4)

    L：cross sectional dimension of the crystalline solid.

    D_v：vacancy diffusion coefficient

                                (9.4.5)

    ΔH_m：enthalpy required to move the vacancy.

• t is in a scale of hours if a solid is heated to about its melting point. (D_v～10^-12m²/s , L～0.1mm)
• Cooling a crystalline solid rapidly (quenching) freezes in an imperfection concentration that is in excess of the concentration in thermodynamic equilibrium.

→ Point imperfections were trapped inside the crystal. They may aggregate together and condense inside the crystal, forming internal voids or a single planner sheet of vacancies. 
 
 
 


9.4.1.2 Point Imperfections in Ionic Solid Are Charged

• A vacancy created by the removal of a positive cation (or anion) results in an effective negative charge ( or positive charge).
• As a result, ionic solids exhibit point imperfection pairs.

Frenkel imperfection (Fig 9.12a)：the positive charged cation interstitials are balanced by the negatively charged cation vacancy left behind.

Schottky imperfection (Fig9.12b)：formed by cation and anion vacancies. The negatively charged cation vacancy and the positively charged anion vacancy are paired to maintain charge balance.

• The concentration of point imperfections, [n] , in ionic solid：

               ----- Frenkel

                ----- Schottky       (9.4.6)

ΔH_fF , ΔH_fS：the enthalpy required to form the Frenkel interstitial – vacancy pair, and to form the Schottky vacancy – vacancy pair.

• Generation in pairs accounts for the factor 2 in the exponential denominator.
• The values for ΔH_fF and ΔH_fS： 
   
   
   
  
• ΔH_fF ＞ΔH_fS
• Depend on the valence of ion being displaced, and the

    crystal structure

• Ex：10 eV for interstitial – vacancy pair (ΔH_fF) in MgO            

           2 eV for vacancy pair (ΔH_fS) in NaCl.

• For ΔH_f = 2 eV.  (temperature effect)

                       at 200℃

                        at 600℃

                        at 1000℃

• [ n ] is extremely sensitive to T and ΔH_f

9.4.1.3 Ionic Solid Surfaces Can Become Charged due to Unequal Formation of Cation and Anion Vacancies

• Because cations are smaller, cation vacancies usually have smaller formation energy than anion vacancies.

       → more cation vacancies (interstitial ?) are created at the surface,

       → surface becomes positively charged.

• The surface charge is compensated by an excess concentration of cation vacancies (negatively charged) just beneath the surface, forming a negative space charge.
• The electrical potential in the space charge beneath the crystal surface follows Gouy – Chapman relationship (Eq 4.3.6) derived for the concentration of counter – ions in the vicinity of a charged surface. (Fig 9.13a)
 


Figure 9.13 Difference in cation and anion vacancy concentration leads to surface charge and compensating internal space change. (a) Pure NaCl；cation vacancies from more readily than anion vacancies. Positively charged surface is compensated by a negative space charge due to excess cation vacancy concentration in surface region. (b) Impure NaCl；containing aliovalent cation impurity (Ca²⁺) of valence greater than sodium . Negatively charged surface is compensated by excess impurity and excess anion vacancy concentration in surface region. 
 


• Impurity ions with different valance from the host lattice may exhibit an effective charge. (Ex：Ca⁺² in NaCl , in Fig 9.13b)
• In this case, the electrostatic potential adds to the chemical potential to enhance the driving force for segregation of charged impurities to the surface.
 


9.4.2 Linear Imperfections, Known as Dislocation Lines, Are Always Present in Crystalline Solids

• Dislocations in crystalline solids are easily introduced by thermal / mechanical stresses developed during crystal growth. The dislocations can rearrange to form small-angle boundaries ( subgrain boundaries )

9.4.2.1 Dislocation Lines are Defined by the Burger Vector

• Figure 9.14a ----- A permanent offset (plastic deformation ) block over the lower half. (p496)
• The strain associated with plastic deformation is not recovered when the load is released.
• Edge dislocation：

Similar to the insertion of an extra half-plane of material into the crystal. The line where this extra half-plane interests the slip plane is called the dislocation lines. ( 9.14a , 9.15a )

• Screw Dislocation ( 9.14b , 9.15b)

A screw dislocation is constructed by distorting the crystallographic plane to form a spiral rather then a flat planar surface.

• Burgers vector of the dislocation line：

For edge dislocation: perpendicular to the dislocation line

For screw dislocation: parallel to the dislocation line. 
 


Burgers Vector

• Dislocations are characterized by a vector called the Burgers vector, b.
• To find the magnitude of direction of b, it is necessary to make an imaginary atom-to-atom circuit around the dislocation, involving one unit of translation in each direction. (attached figure)
• In normal region----the circuit ABCDA is a closed loop and the starting point and finishing point are he same (A).
• Around the dislocation--- 12345—the circuit is not a close loop because 1 and 5 do not coincide.
• The magnitude of the Burgers vector is given by the distance 1-5 and its direction by the direction 1-5 (or 5-1).
 
 


 
 
 
 
 
 


Figure 9.15 Line imperfections in crystalline solids. (a) The configuration of atoms in the vicinity of an edge dislocation. The Burgers vector b is perpendicular to the edge dislocation line AD. Dislocation due to the extra half – plane ABCD leads to compression above the slip plane EFGH and extension below EFGH. (b) The configuration of atoms in the vicinity of a screw dislocation. The Burgers vector b is parallel to the screw dislocation line AD. Dislocation leads to elastic shear strain about the screw dislocation axis. 
 
 
 
 
 


9.4.2.2 Dislocation Lines Possess a High Strain Energy per Unit Length；They Do Not Exist in Thermodynamic Equilibrium

The strain energy per unit length (U) of screw dislocation in a crystal of dimension L

                            (9.4.9)

  G：shear modulus

  b：magnitude of Burgers vector

L_core：dimension of the dislocation core, where the dislocation cannot be described by normal elasticity theory, and the energy required by this dislocation is represented by U_core (U_core is small in comparison with 1^st term )

• For edge dislocation

                            (9.4.10)

                 ν：Poisson’s ratio

• The stain energy required to form dislocations is greater than 10 ev per atomic length of line.

       ( 4 × 10¹⁰ eV/m or 64 × 10^-10 J/m ).

• Such high values preclude dislocations for existing in thermodynamic equilibrium.
• The dislocations are always introduced into crystals during preparation ----- in solidification process due to stresses from mechanical vibrations and thermal expansion mismatch.

9.4.3 Small - Angle Boundaries are Planar Imperfections；They Are Formed from Arrays of Dislocations

• Planar imperfections：small – angle boundaries, stacking faults, twins.
• Small – angle boundaries：A boundary separates small blocks of otherwise perfect crystal. Across the boundary, a small angle of misorientation can be resolved into two compounds. A series of pure edge dislocations spaced equally one above the other.

        ----- A small – angle tilt boundary. 


Figure 9.16 planar imperfections in crystalline solid. (a) A small – angle tilt boundary consists of a vertical array of edge dislocations.

• The angular misorientation θ is given by

                                               (9.4.11)

b：Burgers vector,      

h：vertical spacing between each edge dislocation.

• A small – angle twist boundary may be constructed with a cross grid array of pure screw dislocations.   (Fig 9.16b) , p502
• In general, a small – angle boundary consists of a mixture of edge and screw dislocations. The equilibrium configuration is an interconnected hexagonal “chicken wire” mesh that extends through the solids. ( Fig 9.16c )
• Small – angle boundaries can scatter electrons and modify the behavior of semiconductor devices, act as the source or sink for point defects and impurities in ionic solids. 
  

9.4.3.1 Small – Angle Boundaries Have a Surface Energy Whose Value Derives from the Strain Energy of the Dislocations That Form Them

• The total energy can be computed by summing the strain energies of the constituent dislocation.

The Read – Shockley relationship for the small – angle grain boundary energy per unit area ( γ_sgb ) as a function of small tilt angle θ is：

                             (9.4.12)

   ，   
 
 
 
 
 


• The curve fir eq.(9.4.12) ----- Fig 9.17
 


Figure 9.17 Plot of small – angle grain boundary energy as a function of misorientation 


• For small misorientations (θ), γ_sgb increases linearly, At above 15^o, the γ_sgb approaches a constant value asymptotically ( ～500 mJ / m² , 3×10¹⁸ eV / m² )
 


9.4.3.2 Stacking Faults and Twins Are Other Planar Imperfections – They Have a Low Surface Energy

Stacking faults form when the sequence of planes that make up the crystal structure is disrupted.

Ex：for the {111} planes in face – centered cubic (fcc) structure.

    ‥．a‧b‧c‧a‧b‧c‥． →‥．a‧| b‧c‧b |‧a‧b‧c‥．

    One layer hcp structure | b‧c‧b | appears as a stacking fault. 


Figure 9.18 Planar imperfections in crystalline solids. (a) Normal plane stacking sequence in face – centered cubic crystal structure, ‧a‧b‧c. (b) Stacking fault sequence‧b‧c‧b‧. (c) Twin boundary at a.

• Stacking faults commonly result from crystal growth or plastic deformation process. They also can form from the condensation of excess vacancies into a planar sheet, then one of the layers is missing.

‥．a‧b‧c‧a‧b‧c‥． →‥．a‧| b‧c‧b |‧c‥．

(one layer of hcp structure appears at | b‧c‧b | )

• The stacking fault energy per unit area ( γ_sf ) are about an order of magnitude less than ( γ_sgb：small – angle grain boundary, ~500 mJ/m²)

        160 mJ / m²  within aluminum

        16 mJ / m²  within silver

• Twins form when the stacking sequence is permanently switched in the manner‥．a‧b‧c‧| a‧c‧b‧a‧c‥．across plane | a that mirrors the structure. ( Fig9.18c)
• The twin – boundary energy is one – half the stacking fault energy.

9.5 Kinetic Transport Mechanisms Are Key to Formation, Processing, and Stability of Interfaces in Solid – Solid Systems 


9.5.1 Self – Diffusion and Solute Diffusion in Substitutional Solid Solution by the Vacancy Mechanism

• Each atom vibrates constantly with a frequency ν₀ of about 10¹³ Hz. The symmetry and amplitude of the oscillation depends very sensitively on the bonding character and on the temperature. ( amplitude is ～1Е at T_m)
• When an atom jumps from its lattice location into an adjacent vacant site, the atom and the vacancy exchange locations (Fig.9.19 ). Extra energy is required for this movement.
 


Figure 9.19 Diffusion by vacancy movement. Curve shows the free energy variation as the ion moves through the saddle point in the crystal structure.

• Assuming vacancy diffusion to be a thermally activated process, the atomic jump frequency (ν) is given by

                                    (9.5.1)

ΔG_m：the maximum extra free energy required for the vacancy migration mechanism

  ν₀：atom vibration frequency ～10¹³ per second

• for gold (ΔG_m～1eV , T_m = 1336 K )

        ν～10⁹ per second, means 1 in 10⁴ oscillations

        results in an exchange of position with the vacancy.

• ν～10⁵ per second at room temperature
• The movement may be random without net directional migration.
• At the presence of a potential gradient (Fig 9.20 ), the ΔG for the atom moving down the gradient is less than the ΔG for moving up the gradient. ----- The migration is no longer random, the atoms move down the gradient ( the vacancies move up the gradient ) 
  
• The origins of the potential gradient：chemical, electrical, mechanical, or thermal. We consider the chemical potential gradient here,  
   
   
  
• The net decrease in free energy (χ) per solute atom for forward movement by one atomic spacing (λ)：

                                   (9.5.2)

• The extra free energy needed for the solute atom to pass through the saddle point is：

                     ----- down the gradient

                     ----- up the gradient

  The net jump frequency is given by：

 

   where , when x << 1  →

   So, as χ<< kT ,  
 
 
 


• The net velocity, V_net, of solute atoms associated with this jump frequency is：

                               (9.5.6)

• The net flux of solute atoms, J ( molecule / area‧time)

                          (9.5.7)

     [n]_v：concentration of vacancies ( measured as a fraction )

     c^*：concentration of solute molecules ( molecules per m³)

• Substituting x from eq.9.5.2 into eq.9.5.5 and hence v_net into eq.9.5.7 gives

                             (9.5.8)

• For an ideal dilute solution
 


        So that,

,

                              (9.5.9) 
 
 
 
 


• Substitution     ΔG_m = ΔH_m + TΔS_m  ,

                              (eq.9.4.3)

        We obtain

                       (9.5.10)

A：contains configurational entropy terms associated with both the formation, A’ , and movement, A” , of the vacancy.

• Comparing with Fick’s first law, the diffusion coefficient for the substitutional solute atom, Ds, is：

              or

                                         (9.5.11)

                D₀ = v₀λ²A ； ΔH_d = ( ΔH_f + ΔH_m )

• For an ideal solution in which the size and chemical character of the solute and solvent atoms are identical, Ds equals the self – diffusion coefficient ( D_self ) ----- the diffusion of solvent atom in the solvent itself.
• For nonideal solutions, D_s depends on how the value of ΔH_m is changed by misfit strain (ε_m), elastic modulus (E), , and chemical interactions.
• The activation energy for vacancy migration is less than the activation energy for atom diffusion (because a vacancy always has an adjacent atomic site that it is able to move into). D_v is always larger than D_self. The enthalpy is only ΔH_m. 
  
• Since , Eq( 9.5.11 or 12 ) becomes
• Diffusion in ionic solid： 
  

       Because the anion is generally larger than the cation, it requires greater energy to squeeze through the saddle point.

• Typical values for D₀ and ΔH_d ( Table9.3 )

        Table 9.3a ----- for pure cubic metals 


• The self – diffusion coefficient of metals：

     ~ 10^-13 m² / s               just below the melting point

     ~ 10^-9 m² / s                just above the melting point

     10^-18 ~ 10^-24 m² / s         at room temperature

• Diffusion is most difficult in covalently bounded solids, because of the need to break and remake the covalent bonds.
 


9.5.2 Self – Diffusion and Solute Diffusion in Interstitial Solid Solutions by the Interstitial Mechanism

• If the solute atom is much smaller than the solvent atom (＜60％), then the solute atom prefers to locate at interstitial sites to minimize mechanical distortion.
• Solute migration then occurs rather easily by jumping from one interstitial site to another ( Fig 9.21a )

         (enthalpy for solute migration ) 


Figure 9.21 Diffusion by interstitial movement. (a) An impurity atom much smaller than the host diffuses through the solid by jumping from an interstitial site (1) into an adjacent interstitial site (2). 


• If all the atoms are the same size, the interstitial mechanism is a more formidable mode of self – diffusion due to the large strain energy required to form the interstitial.
• Once formed, an interstitial can move by the interstitialcy mechanism ( Fig 9.21b ) ----- a two – step mechanism.
 
 


Figure 9.21 (b) Diffusion by the interstitialcy mechanism. An atom moving in from the left displaces atom (1) into an interstitial site. Atom (1) then displaces atom (2) resulting in net diffusion through the crystal structure. 


• Interstitial diffusion mechanism are more common in ionic solids because the two elements are of different size and the crystal structure is less compact.

           ( ΔH_fF：the energy to form Frenkel defect )

• In some ionic solids, the preferred diffusion mechanism ( vacancy or interstitial ) is different for cations and anions.

       Ex：In AgCl：

        Ag^＋：move via interstitialcy mechanism

        Cl^－( larger )：via the vacancy mechanism. 


3. Diffusion and Absolute Mobility Are Related by the Nernst – Einstein Equation
•              v = F B                          (9.5.14)

  B：the absolute mobility of an atom

  F：the force driving an atom

  v：the velocity of the atom

• the driving force for diffusion, F_diff, is

                                    (9.5.15)

The flux J is given by：

                                         (9.5.16)

and                                      (9.5.17)

• From 9.5.14 →

                                    (9.5.18)

This is the Nernst – Einstein equation relating mobility and diffusivity. 
 
 
 
 


• The friction factor f and the diffusion coefficient (D) for a particle passing through a fluid ( eq 3.8.35)
 


is equivalent to the absolute mobility, B

• A linear relation exists between a potential gradient and the flux J ,

K：transport coefficient

：potential gradient , can be

      chemical：diffusion of atoms

      electrical：ion conductivity

      thermal：thermal migration

4. Fick’s Second Law Describes Changes in Composition with Time

                                         (9.5.19)

Consider the penetration of a surface coating into the interior of a semi – infinite solid：

      B.C.      C＝C_s       at  x＝0    for   0< t< ∞

      I. C.      C＝C₀      at  t＝0    for   0< x< ∞

The solution is       
 


• D_s：the diffusion coefficient for the diffusion of surface coating elements ( s ) in the bulk material. The value depends on：temperature, element, the strain, the diffusion mechanism. Assumption：D_s does not vary with composition
• At the depth the bulk composition , x is termed x_0.5, then
 


    The solution for x_0.5 is approximately：

             (9.5.21)

• The penetration depth is given by, within 1 sec, x_0.5 or about 10^-2 cm ( D = 10^-8 m² / s) , 10^-1μm ( D = 10^-14 m² / s) , and 1Е ( D = 10^-20 m² / s)
• Two important points are highlight.
1. The stability of an interface is extremely sensitive to temperature.
2. the stability of an interface depends on the scale of our concerns
• Since , . Means that the time for diffusion to a fixed depth x is proportional to .

Ex：for movement of 1μm, the time are about 10^-4s, 100s, and 10⁸s ( 3years ) for, respectively, diffusion coefficient of 10^-8, 10^-14, and 10^-20 m² / s

• Interdiffusion of two solids of composition c₁ and c₂.

IC﹒   c = c₁  for  x < 0     , at t = 0

       c = c₂  for  x > 0     , at t = 0

BC﹒  c = c₁  for   x = - ∞

       c = c₂  for   x = ∞  ,

the solution is  


Assumption：D_s does not vary with composition 


• at x = 0 , , keeps constant
• the extent of interdiffusion is also proportional to
5. Large Difference in Diffusivity Lead to Movement of Interfacial Boundaries with Time and the Formation of Interfacial Defects
• When D does depend on composition
 


    D_s2：diffusion coefficient of solute 1 in solvent 2 ( D₁ )

    D_s1：diffusion coefficient of solute 2 in solvent 1 ( D₂ )

    X₁：atomic fraction of solute 1

    X₂：atomic fraction of solute 2

• If solute 2 is very dilute in solvent 1

         , , →

It means that the diffusion coefficient of the solute ( D₂ ) control the interdiffusion behavior.

• When D1 and D2 are very different, a net mass transfer occurs across the original interface, and the interface appears to move.
• Example：the move of interface between brass ( Zn + Cu ) and copper

Zn diffuses (by the interstitial mechanism ) at a much faster rate than the copper diffuses ( by the substitutional vacancy mechanism ). The number of Zn atoms moving out of the brass far exceed the number of copper atoms moving in to replace them. → the brass/ copper interface move into the copper, and a net excess of vacancies left behind in the brass. The vacancies can adsorb on dislocation lines (low concentration ) or condense as macroscopic spherical voids ( high concentration ) 
 
 
 
 
 
 
 
 


6. Diffusion over Surfaces Is Faster Than through the Bulk of the Material
• The energy required for an interstitial atom to migrate is lower along the dislocation line, grain boundary, and over a free surface.
• Diffusion along dislocation line is referred to as pipe diffusion   Diffusion along grain boundary is referred to grain boundary diffusion                                      Diffusion over free surface → surface diffusion
• Δ H_lattice >Δ H_pipe >Δ H_gb >Δ H_surf                                 D_lattice << D_pipe << D_gb << D_surf                     (9.5.24)
• Typically, we can assume that                            Δ H_lattice：Δ H_gb：Δ H_surf ～ 4：2：1                        D_os range between 10^-5 and 10^-4 m² / s
• D_surf is approximately five orders of magnitude larger than the D_lattice at .      
• At moderate temperature, () , D_gb can be ten orders of magnitude larger than the D_lattice .
7. Material Transport in Solid Also can be Driven by Electrical Potential Gradients and Mechanical Strain Gradients