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

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)