Surface Energy and Surface Stress in Phase-Field Models of Elasticity

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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
• 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
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Surface Excess Quantities (Gibbs)
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Kramers Potential (fluid system)
(surface energy)
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1-D Elastic System (single component)
z

Liquid
Solid
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Kramers Potential (elastic system)
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Planar Geometry
z

• Solid and liquid separated by an interface
• Planar geometry
• No dynamics
• Applied uniaxial stress or biaxial strain
• ? 1D problem

t0

Liquid
eS
Solid

• Examine
• Equilibrium temperature (T0)
• Surface energy and surface stress (Gibbs
  adsorption)
• Analytical results and numerical results are
  compared

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Phase-Field Model of Elasticity

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1-D Phase-Field Solution
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1-D Stress and Strain Fields
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Analytical Results Melting Temperature

• Here is the analytical results for the melting
  temperature. Using the first integral of the
  system, we obtain the constant quantity that
  gives equilibrium. Here, f is the free energy
  including the temperature effect, fel is the
  elastic energy density, this is the gradient
  energy, and the last term is the work term due to
  applied stress, where tau0 is the applied stress,
  U is the strain ezz. Using analytic solution
  for the elastic equilibrium, we obtain this
  expression for the melting temperature shift due
  to the presence of applied stress or applied
  strain. Note that the shift depends only on the
  jump of the quantities, and does not depend on
  the interfacial thickness.

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Numerical Simulation Melting Temperature

• For numerical simulation, we take a physical
  parameter for Aluminum eutectic (or an estimate
  for it. The equations are non-dimensionalized
  using the latent heat per unit volume and the
  system size. We now focus on the case with
  applied stress only, that is tau0 is non zero,
  and msifit and applied strain is zero. Then the
  expression for the melting temperautre change
  becomes this. The change is proportional to the
  applied stress2.

• 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
  

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Simulation and analytics agree

• Indeed the simulation results agree with the
  analytical results.

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Analytical Results Surface Energy

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Numerical and analytical results agree

• Here, I plot dgama dtau0 agaist tau0. The
  numerical solution agrees with the analytical
  solution for dgamma dtau0.

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Elastic Equilibrium of a Spherical Inclusion
Bulk modulus, KLKSK
Shear modulus, m0 in liquid
VSltVL Self-strain e0djk in liquid
0 in solid
(1)
(2)
R1
R
DffS-fL LV (T-T0)/T0
Compare phase-field sharp interface results for
Claussius-Clapyron/Gibbs-Thomson effects
numerics asymptotics

Johnson (2001)
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Phase-Field Model
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Sharp-Interface Model
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Interface Conditions
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Solid Inclusion
S
L
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Liquid Inclusion
L
S
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Phase-Field Calculations
Liquid fraction
S
L
T/T0
Liquid-Solid volume mismatch produces stress and
alters equilibrium temperature (Claussius-Clapyron
)
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Phase Field vs Sharp Interface (no surface energy)
Liquid fraction
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Phase Field vs Sharp Interface (surface energy
fit)
Liquid fraction
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Conclusions

• 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

Future Work

• More detailed numerical evaluation of surface
  stress in 3-D
• Derive formal sharp-interface limit of
  phase-field model

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(End)