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Surface Energy and Surface Stress in Phase-Field Models of Elasticity. J. Slutsker , G. McFadden, J. Warren, W. Boettinger, (NIST). K. Thornton , A. Roytburd, P. Voorhees, (U Mich, U Md, NWU). Surface excess quantities and phase-field models - PowerPoint PPT Presentation
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J. SlutskerJ. Slutsker, G. McFadden, J. Warren, W. Boettinger, (NIST), G. McFadden, J. Warren, W. Boettinger, (NIST)
K. ThorntonK. Thornton, A. Roytburd, P. Voorhees, (U Mich, U Md, NWU), A. Roytburd, P. Voorhees, (U Mich, U Md, NWU)
Surface Energy and Surface Stress in Phase-Field Models of Elasticity
•Surface excess quantities and phase-field models
•1-D Elastic equilibrium – axial stress & biaxial strain
•3-D Equilibrium of two-phase spherical systems
Goal: illuminate phase-field description of surface energy and surface strain by simple examples
Surface Excess Quantities (Gibbs)
Kramer’s Potential (fluid system)
(surface energy)
z
Solid
“Liquid”
1-D Elastic System (single component)
“Kramer’s Potential” (elastic system)
Planar Geometry
•Solid and “liquid” separated by an interface
•Planar geometry
•No dynamics
•Applied uniaxial stress or biaxial strain
1D problem
0
z
Solid
Liquid
•Examine
Equilibrium temperature (T0)
Surface energy and surface stress (Gibbs adsorption)
•Analytical results and numerical results are compared
eS
Phase-Field Model of Elasticity
1.0
0.8
0.6
0.4
0.2
0.0
1.00.80.60.40.20.0
0.06
0.05
0.04
0.03
0.02
0.01
0.00
1-D Phase-Field Solution
1-D Stress and Strain Fields
Analytical Results: Melting Temperature
• First integral
•We thus obtain,
where denotes the jump across the interface
Numerical Simulation: Melting Temperature
• “Physical” parameters for Aluminum eutectic is used
• Variables are non-dimensionalized using the latent heat per unit volume and the system length
• Here, we focus on applied stress with no misfit:
Simulation and analytics agree
Analytical Results: Surface Energy
• Surface energy is associated with the surface excess of thermodynamic potential [Johnson (2000)]
• “Gibbs adsorption equation” can be derived [Cahn (1979)]:
Numerical and analytical results agree
L SuS=0
T
Bulk modulus, KL=KS=K
Shear modulus, =0 in “liquid”
VS<VL
Self-strain: jk in liquid 0 in solid
R1
R
f=fS-fL= LV (T-T0)/T0
(1) (2)
Compare phase-field & sharp interface results for Claussius-Clapyron/Gibbs-Thomson effects [numerics & asymptotics] [Johnson (2001)]
Elastic Equilibrium of a Spherical Inclusion
Phase-Field Model
Sharp-Interface Model
Interface Conditions
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
0 100 200 300 400 500 600 700 800 900 1000
LS
Solid Inclusion
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
6.00E-01
0 100 200 300 400 500 600 700 800 900 1000
L S
Liquid Inclusion
0
0.2
0.4
0.6
0.8
1
1.2
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
T/T0
Liq
uid
frac
tion
S
L
Phase-Field Calculations
Liquid-Solid volume mismatch produces stress and alters equilibrium temperature (Claussius-Clapyron)
0
0.2
0.4
0.6
0.8
1
1.2
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Phase Field vs Sharp Interface (no surface energy)L
iqui
d fr
acti
on
T/T0
0
0.2
0.4
0.6
0.8
1
1.2
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Phase Field vs Sharp Interface (surface energy fit)L
iqui
d fr
acti
on
Conclusions
Future Work
• Phase-field models provide natural surface excess quantities
• Surface stress is included – but sensitive to interpolation through the interface
• Surface energy and Clausius-Clapyron effects included
• More detailed numerical evaluation of surface stress in 3-D
• Derive formal sharp-interface limit of phase-field model
(End)