EMA5001 Lecture 9 Surface Energy - FIU Mechanical And .
EMA5001 Lecture 9Surface Energy 2016 by Zhe Cheng
Types of Interfaces for Solids By the materials on each side of the interface Surface Between solid and vapor (or vacuum) Always present Matters in vaporization, condensation, wetting, fracture, etc. Grain boundaries Separate solids with same composition and same crystal structure but differentorientation Can be absent Matters in recrystallization, grain growth, mechanical strength, hardness, etc. Interphase interfaces Separate two phases with different crystal structure and/or compositions Matters in phase transformation, mechanical properties, etc. Most complex and least understoodEMA 5001 Physical Properties of MaterialsZhe Cheng (2016)9 Surface Energy2
Interfacial Free Energy & Surface Tension For a system with interface area A and free energy per unit interfacearea , the total free energy isG G0 A G0 is free energy of system assuming all material has the bulk properties reflects the excess free energy of material on surface, equaling the work thatmust be done at constant T and P to create unit area of surface from bulkmaterial At constant T and P, if interface area increase by dA, system free energydG dA Ad change For liquidF is surface tensionTherefore,For liquid,d 0dAdG dW FdAd F AdAEMA 5001 Physical Properties of MaterialsF Zhe Cheng (2016)9 Surface Energy3
Energy of a Particular Surface Plane (1) Example of (111) plane as surface of FCC crystal Bond strength1If a bond is broken, the energy of one of the atom is raised by2Compared with bulk, every surface atomon (111) surface plane has lost three (3)of the nearest neighboring atoms Excess internal energy for each of the (111) 3 1 2surface atom over an atom in the bulk is If LS is the measured sublimation energyLS 12 N aper mole, For FCC structure,Na is Avogadro’s number. Therefore LS / 6 N a (111) surface (solid-vapor interface) excessEsv (111)internal energy per surface atom is Assumptions: 1 2 L33 L S S22 6Na4Na Consider only nearest neighbors Strength of remaining bonds unchangedEMA 5001 Physical Properties of MaterialsZhe Cheng (2016)9 Surface Energy4
Energy of a Particular Surface Plane (2) Continue from p.4For FCCExcess energy per atom on a (111) surface planeEsv (111) L33 L S S22 6Na4NarExcess surface energy per unit area (if neglecting entropy term) (111) Esv n(111)2r n is atom number density on (111) planen(111) 0.51 20.5 2r 3r2 3rTherefore, (111) Esv n(111) 1 2 2 3 a 4 2 2a24 1 23 aLSLSLS4 11 2 0.5824NaNaa23 a3 NaaEMA 5001 Physical Properties of MaterialsZhe Cheng (2016)9 Surface Energy5
Energy of a Particular Surface Plane (3) Surface energy of other planes Broken bond excess energy / 2 We have LS / 6 N aAtom on (002) surface plane lost four (4)of the nearest neighborsL 4 LEsv (002) 4 S S2 2 6 N a 3N aLLS12 LS (002) S 0.673N a 2 2 3 N a a 2N a a 2 http://www.tf.uni-kiel.de/matwis/amat/def en/kap 1/illustr/t1 3 3.html a 2 Surface energy (excess energyAtom on (220) surface plane lost five (5)with respect to bulk) changeof the nearest neighbors5 LS 5 Lwith different crystal planesEsv (220) 5 S 2 2 6 N a 12 N aLSLS5 LS15 (220) 0.59212 N aNaa226 2 Naaa a2EMA 5001 Physical Properties of MaterialsZhe Cheng (2016)9 Surface Energy6
Surface Energy & Surface Orientation (1) y [010]x [100]Number of broken bondsalong (001) planeCos 1 aaaNumber of broken bondsalong (100) planeSin Sin /a broken bonds If a 3D crystal surface atan angle to close packed(001) plane for a simplecubic lattice structure z [001]a1 aCos /a broken bondsEnergy associated with each broken bond / 2Total excess surface internal energy (with respect to the bulk) per unit areaEsv ( ) EMA 5001 Physical Properties of MaterialsCos Sin a2Zhe Cheng (2016) 29 Surface Energy7
Surface Energy & Surface Orientation (2) Continue from p.7Esv ( ) Cos Sin Plotting ESV vs. a2 ESV 2Closed packed orientation lie atenergy minimum (cusp or trough) positions))0 Example of γ in polar coordination system for simple cubic lattice angle to close packed {100} type planes, for 0 90o 11 2 Cos Sin 2 Cos Sin 22a2a22 Cos 4 2a 2EMA 5001 Physical Properties of MaterialsZhe Cheng (2016)9 Surface Energy8
Wulff Planes & Equilibrium ShapeContour of γ plot Continue from p.7 Cos 4 45oWulff planes(010) 2a 2A circle off byand diameter of Plotting γ ( ) in polar systemEsv (110) 2a 2Esv (100)(100)As function of and absolute value ofsurface excess energy, yielding schematic Equilibrium shape of a crystal determined by Minimization of the total surface energy iEquilibriumshapeAiWulff Theorem“The equilibrium shape is bonded by those parts of Wulff planes that can bereached from the origin without crossing any other Wulff planes”J. W. Martin, et al. Stability of microstructure in metallic systems, Cambridge Univ.Press, 2nd ed, Cambridge, UK (1997), p. 234EMA 5001 Physical Properties of MaterialsZhe Cheng (2016)9 Surface Energy9
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 9 Surface Energy Interfacial Free Energy & Surface Tension For a system with interface area A and free energy per unit interface area , the total free energy is G 0 is free energy of system assuming all material has the bulk properties
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EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 4 Self-Diffusion & Vacancy Diffusion Diffusion of Vacancy vs. Substitutional Atoms Continue from p. 7 2 Therefore, Diffusion coefficient of vacancy vs. substitutional atom For self-diffusion 2 The relationship between jump frequency is Since the jump distance is the same
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a central part of the Revolution’s narrative, the American Revolution would have never occurred nor followed the course that we know now without the ideas, dreams, and blood spilled by American patriots whose names are not recorded alongside Washington, Jefferson, and Adams in history books. The Road to the War for American Independence By the time the first shots were fired in the American .
