Phase Field Models Microstructures and Their Evolution T.A. Abinandanan Department of Materials Engineering Indian Institute of Science Bangalore J. A. Krumhansl School on Unifying Concepts in Materials JNCASR-January 2012 Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 1 / 43
4’ 33" 4’ 33” is a ... composition by American experimental composer John Cage (1912 - 92). It was composed in 1952 for any instrument (or combination of instruments). The score instructs the performer not to play the instrument during the entire duration of the piece. Source: Wikipedia Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 2 / 43
Monochrome Art Yves Klein: IKB 79 Source: Tate Online www.tate.org.uk Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 3 / 43
Monochrome Art Yves Klein: IKB 81 Source: Yves Klein Archives www.yveskleinarchives.org/ Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 4 / 43
Art with "microstructure" Hans Hofmann Source: Photos of Abstract Art www.photosofabstractart.com/ Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 5 / 43
Microstructural "art" Xi-Ya Fang, Monash University Source: Monash Universityt mcem.monash.edu.au/assets/images/gallery/7001f3-09.jpg Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 6 / 43
Art with "microstructure" bterrycompton on Flickr farm8.staticflickr.com/7023/6708508023_913b35b086.jpg Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 7 / 43
Microstructural "art" Vikram Jayaram et al, IISc materials.iisc.ernet.in/ qjayaram/Mechanicalproperties.htm Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 8 / 43
Microstructures are everywhere Optical microscopy: 1 - 100 µ m. Electron microscopy: Down to 10 nm. Mesoscale: Large compared to atomic sizes and small compared to what our eyes can see. Computational modeling has contributed much to our understanding of how they form and evolve. Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 9 / 43
Types of models Continuum theories : emphasis on analytical solutions 1 Computational models 2 Atomistic models : Molecular dynamics, Monte Carlo Continuum models : Phase field models Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 10 / 43
Early continuum models Emphasis on Analytical solutions. Isolated features . Specifically, their shape , or the dominant length scale. Spherical, ellipsoidal, plate-like or rod-like particles Dendrites: Alternating lamellae: lamellar spacing Spinodal: Maximally growing composition fluctuation. Studies of evolution of multiple features were restricted to simple shapes e.g. Lifshitz-Slyozov-Wagner theory of coarsening of spherical particles. Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 11 / 43
Spinodal microstructures A. Chiasera et al (SPIE Newsroom, 2011) Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 12 / 43
Au-Ag de-alloying Erlebacher et al (2001) Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 13 / 43
Porous Glass M. Suzuki et al, J. Phys. Conf. Ser. (2009) Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 14 / 43
Miscibility Gap Phase diagram of 2,6-lutidine-water system E.Herzig, University of Edinburgh PhD Thesis (2008), Nature Materials (2007) Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 15 / 43
Ising model: Monte Carlo simulations Magnet: Salvatore Torquato, Phys. Bio. (2011) - Magnet Alloy: Iyad Obeid (obeidlab.blogspot.com) Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 16 / 43
Miscibility Gap Nearest neighbour bonds e AA , e BB and e AB Miscibiligy gap when e AB > ( e AA + e BB ) / 2 High Temperature: homogeneous alloys Low temperature: Co-existence of A -rich and B -rich phases Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 17 / 43
Ising model: Monte Carlo simulations Dave Johnson Web applet in HTML 5 (dtjohnson.net/projects/ising) Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 18 / 43
Coarse graining Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 19 / 43
Compositionally diffuse Interface Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 20 / 43
Free energy of non-uniform system Cahn-Hilliard Model � � f o ( c )+ κ | ∇ c | 2 � F = N v dV V F = The total free energy of the system f o [ c ( x )] = Free energy density at location x (depends on the local composition) κ = Gradient energy coefficient ... Let’s look at the details. Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 21 / 43
Free energy functional Start with: � F = N v V f ( r ) dV Free energy per molecule, f ( r ) , must depend not only on the local composition, but also on composition derivatives. f ( r ) = f ( c , p i , q ij ... ) p i = ∂ c / ∂ x i ∂ 2 c / ∂ x i ∂ x j q ij = Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 22 / 43
Taylor series expansion of f ( c , p i , q ij ) q ij + 1 f ( c , p i , q ij ) = f o ( c )+ L i p i + κ ( 1 ) 2 κ ( 2 ) p i p j + ... ij ij f o ( c ) = f ( c = c , p i = 0 , q ij = 0 ,... ) � ∂ f � L i = ∂ p i ( c , 0 , 0 ) � ∂ f � κ ( 1 ) = ij ∂ q ij ( c , 0 , 0 ) � ∂ 2 f � κ ( 2 ) = ij ∂ p i ∂ p j ( c , 0 , 0 ) Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 23 / 43
Symmetry arguments Centre of symmetry: energy due to p i and − p i must be the same. L i = 0; Cubic symmetry (or, isotropy): κ ( 1 ) = κ 1 δ ij ij κ ( 2 ) = κ 2 δ ij ij f ( c , p i , q ij ) = f o ( c )+ κ 1 q ii + 1 2 κ 2 p i p i or, f ( c , p i , q ij ) = f o ( c )+ κ 1 ∇ 2 c + 1 2 κ 2 | ∇ c | 2 Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 24 / 43
Divergence theorem � ∂ k 1 � � � [ ∇ c ] 2 + � � � k 1 ∇ 2 c dV = − S [( κ 1 ∇ c ) · n ] dS ∂ c V V We now have the final form for the total free energy, F , for a system with a non-uniform composition field c ( r ) : Cahn-Hilliard Model: Energy � � f o ( c )+ κ | ∇ c | 2 � F = N v dV V with κ = − ∂κ 1 ∂ c + κ 2 2 Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 25 / 43
Equilibrium interface width Cahn-Hilliard Model: Energy � � f o ( c )+ κ | ∇ c | 2 � F = N v dV V Too sharp Just right Too wide κ | ∇ c | 2 is too high f o ( c ) is too high Minimizes F Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 26 / 43
Interfacial energy and width � ∞ � ∆ f o + κ | ∇ c | 2 � σ = N v dx − ∞ where ∆ f o = f o ( c ) − ( 1 − c ) µ A − c µ B Cann and Hilliard show that: σ ∝ [ κ ∆ f o ] 1 / 2 � κ � 1 / 2 w ∝ ∆ f o Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 27 / 43
Diffusion Diffusive flux, J : J = − M ∇ µ Diffusion potential, µ = ( µ B − µ A ) : µ = δ F δ c Atomic mobility, M Continuity equation: ∂ c ∂ t = − ∇ · J = ∇ · [ M ∇ µ ] Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 28 / 43
Cahn-Hilliard equation For constant mobility, M : Cahn-Hilliard Model: Kinetics ∂ c ∂ t = M ∇ 2 µ where: µ = δ F δ c = ∂ f c ∂ c − 2 κ∇ 2 c With composition, c , scaled so that c α = 0 and c β = 1, the simplest approximation for f o ( c ) (double well potential) is: f o ( c ) = A c 2 ( 1 − c ) 2 Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 29 / 43
Spinodal decomposition: Microstructural evolution Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 30 / 43
Simulation procedure Discretize space using a regular lattice (grid). and start with an initial composition field c ( r , t = 0 ) . Typically, a uniform composition (say, c = 0 . 5) with a small random fluctuation added to it. Solve a discretized version of the Cahn-Hilliard equation 1 numerically to go from c ( r , t ) to c ( r , t +∆ t ) . Any suitable numerical procedure will do. We use a Fourier transform technique. Go back to Step 1! 2 Store the composition field every once in a while for post-simulation data analysis and visualization. Abinandanan (Materials Engg, IISc) Phase Field Models J.A. Krumhansl School 31 / 43
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