材料热力学讲义-13

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Effect of Surface Curvature
Molar volume Surface energy
Radius of curvature
1 1 1 1 ( ) r 2 r1 r2
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Vapor Pressure as a Function of Radius
Condensed phase – vapor equilibrium: Valid for both curved and flat surfaces:
Activity
This process is called Coarsening / Ostwald ripening
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Melting Point of Small Particles
Solid particle
Liquid
Assume pressure change in liquid is small
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Various Types of Surfaces / Surface Energies Stacking faults Twin boundaries Domain boundaries Antiphase domain boundaries Coherent interfaces Incoherent interfaces ……
Relative Surface Energy Measurement
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Measurement of Surface Energy
Relative Surface Energy Measurement
Grain boundary grooving:
Example: Ni bicrystal
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Measurement of Surface Energy
Bubble Method for Liquids
Capillary Rise Method for Liquids
y Density
Gravity constant
Radius
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Measurement of Surface Energy
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(creating 2 surfaces)
Surface energy:
Number of atoms per unit area
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Surface Energy
e.g.: Cu
Surface energy:
Number of atoms per unit area
HS = 170 kJ/mole a0 = 0.3615 nm = 1400 ergs/cm2 Experimental: 1600 ergs/cm2
Entropy of melting
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Melting Point of Small Particles
Example: 10 nm particle of Au:
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Melting point of Au as a function of particle diameter
1336K
BJ Lee
(The shape of the liquid droplet can be used to measure the surface energy of the liquid)
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Temperature Coefficient of Surface Energy
At constant P:
Surface entropy per unit area
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Measurement of Surface Energy
Relative Surface Energy Measurement
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Measurement of Surfawk.baidu.come Energy
Surface Wetting Method for Surface Energy Measurement
Number of nearest neighbors Number of broken bonds
Lupis
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Surface Energy vs Surface Tension
dG = – S dT + V dP + dA

Surface Energy:

G A T , P
Lecture 13
Thermodynamics of Surfaces and Interfaces and Effects of External Fields on Thermodynamics
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Surface Energy
• An atom at a free surface of a solid has greater energy than an atom in the interior of a crystal because it is less tightly bound. • The surface atoms, being less bound to others, exist at higher energy levels than atoms in the bulk of the crystal • The sum of all the excess energy of the surface atoms (excess with respect to atoms inside the crystal) is essentially the surface energy
(111) of fcc
Lattice parameter
(100) of fcc
Surface energy is a function of crystallographic orientation
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2
Surface Energy
Molar surface energy of liquid:
Molar enthalpy of evaporation
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Equilibrium Shape of A Crystal
Gibbs-Wulff construction/plot
If you haven’t seen the Smithsonian’s National Natural Museum of History in Washington, DC, you got to go and appreciate the shear beauty of Nature Rock and Gemstone Section
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0
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Surface Energy
Quasi-Chemical Approximation: only counts the nearest neighbor bonds
Molar enthalpy of sublimation (breaking all the bonds) Energy per bond: Avogadro number Coordination number For the (111) surface of fcc, three bonds per atom will be broken Work required to form a surface will be 2 Work per surface atom:
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r1
2
r2
Mean curvature:
1 (1 2 ) ( ) 2
1 1 1 1 ( ) r 2 r1 r2
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1 1 2 r1
1 r2
Mean radius:
(Mechanical Analogy Derivation of the Equation)
Effect of Surface Curvature
fugacity
Assume ideal gas behavior:
Vapor pressure of a spherical particle is a function of its radius. Smaller particles have higher vapor pressure, thus tend to shrink. Larger particles will grow.
SS is generally positive
<0
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Equilibrium Shape of A Crystal
Gibbs-Wulff construction/plot
Isotropic sphere
Area: Total surface energy: Minimize surface energy at constant area When
Surface energy is often called surface tension (It’s the same for liquid, not the same for solids)
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3
Surface Curvature
Curvature
Radius of curvature
For any smooth curved surface, one can always find a perpendicular pair of curvatures, one is the maximum and the other is a minimum – these are called the principal directions. 1 1
Density Gravity constant
Area
is constant (no evaporation)
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Measurement of Surface Energy
Zero Elongation Method for Solids Grain boundary effect
gb = grain boundary Number of grain boundaries per unit length
• Surface energy against vapor • Average values – cannot get dependency of crystallographic orientations • Surface energy against various gases
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Measurement of Surface Energy
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Many Different Types Interfaces
decreasing r
A r decreases, d XB,r increases i As Matrix phase surrounding small particles has higher concentration of B. At high enough temperature when diffusion takes place, small particles will dissolve and larger g p particles will g grow.
H
U H F G A S ,V A S , P A T ,V A T , P
U S V
P T F
G
in dyne-centimeters
Newton/meter ergs/cm2 J/m2
Zero Elongation Method for Solids At T near melting point • Wire shrinks due to surface tension • Wire elongates due to stress imposed by weight • The balance is zero elongation
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Solubility as a Function of Particle Radius
Define standard B as infinite flat pure B:
Assume Henry’s Law:
Activity
As r decreases, XB,r increases
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Solubility as a Function of Particle Radius